0 on an interval, then fis concave up on that interval. We can see that if there is an inflection point it has to be at x = 0. "Here is what helped me: If the sign of the second derivative changes as you pass through the candidate inflection, "Short and to-the-point, with enough detail to cover all the procedures. How do you find inflection points on a graph? For example, to find the inflection points of one would take the the derivative: I used the second derivative to find them but I can't, the second derivative does not cancel its returns null. While I have been able to find critical number, I'm not sure how to find the inflection point for the function as for this particular function I cannot assign double derivative to be zero and then solve for x. Economy & Business Elections. inflection points y = x3 − x. Start with getting the first derivative: f '(x) = 3x 2. And the inflection point is at x = −2/15 Finding Points of Inflection. from being "concave up" to being "concave down" or vice versa. One of these applications has to do with finding inflection points of the graph of a function. Inflection points can be found by taking the second derivative and setting it to equal zero. Finding critical and inflection points from f’x and f”x – What is the top of a curve called? An inflection point is defined as a point on the curve in which the concavity changes. Hint: Enter as 3*x^2 , as 3/5 and as (x+1)/(x-2x^4) To write powers, use ^. I just wanted to find the xval where a more complicated function changes direction in particular ranges that I can iterate over: find_root(diff((x^2)*cos(2*x)),-5,-2) then results in -3.2891668663611693, which corresponds with its graph., that I put in above to clarify. ", "The article makes the problem about inflection points much simpler. This is because linear functions do not change slope (the entire graph has the same slope), so there is no point at which the slope changes. Say you need to find the inflection point of the function below. Example 1 with f( x) = x3. Here, we will learn the steps to find the inflection of a point. How do I determine the dependent and independent variable in a relation or function? Remember that you are looking for sign changes, not evaluating the value. Calculus is the best tool we have available to help us find points of inflection. We can clearly see a change of slope at some given points. The procedure to use the inflection point calculator is as follows: Step 1: Enter the function in the respective input field. To find inflection points of, solve the equation h = 0. Inflection Points by Frederick Kempe. If you really can’t stand to see another ad again, then please consider supporting our work with a contribution to wikiHow. Sun, Dec 6, 2020 Biden’s rare shot at a transformative presidency runs through Europe and China Joe Biden has that rarest of opportunities that history provides: the chance to be a transformative foreign-policy president. Enter the function whose inflection points you want to find. Inflection points are defined where the curve changes direction, and the derivative is equal to zero. View problems. For that equation, it is correct to say x is a function of y, but y is not a function of x. Active 8 months ago. inflection points f ( x) = x4 − x2. Let’s do an example to see what truly occurs. Credits The page is based off the Calculus Refresher by Paul Garrett.Calculus Refresher by Paul Garrett. [1] ", "This article helped me to find out the inflection point of a curve. By signing up you are agreeing to receive emails according to our privacy policy. The point at which the curve begins is the springing or spring-line. f''(x) = 6x^2 + 12x - 18 = 0 . One idea would be to smooth the data by taking moving averages or splines or something and then take the second derivative and look for when it changes sign. Very helpful! Compute the first derivative of function f(x) with respect to x i.e f'(x). References. The geometric meaning of an inflection point is that the graph of the function $$f\left( x \right)$$ passes from one side of the tangent line to the other at this point, i.e. In Mathematics, the inflection point or the point of inflection is defined as a point on the curve at which the concavity of the function changes (i.e.) There are rules you can follow to find derivatives, and we used the "Power Rule": And 6x − 12 is negative up to x = 2, positive from there onwards. For more tips on finding inflection points, like understanding concave up and down functions, read on! How to find a function with a given inflection point? Let's take a look at an example for a function of degree having an inflection point at (1|3): f'(x) = 2x^3 + 6x^2 - 18x. $inflection\:points\:f\left (x\right)=x^4-x^2$. Inflection points are points where the function changes concavity, i.e. Multiplying 6 by -6 will give you a result of -36, not 0. wikiHow is where trusted research and expert knowledge come together. According to the Intermediate Value Theorem, the second derivative can only change sign if it is discontinuous or if it passes through zero, so let's take the second derivative and set it equal to zero. Ask Question Asked 8 months ago. An inflection point gives multiple equations: On the one hand, you got the y-value. This is because an inflection point is where a graph changes from being concave to convex or vice versa. fplot (f, [-9 6]) hold on plot (double (inflec_pt), double (subs (f,inflec_pt)), 'ro') title ('Inflection Point of f') text (-7,1, 'Inflection point') hold off In differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a continuous plane curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. Take any function f(x). If the sign does not change, then there exists no inflection point. Decoding inflection points past, present, and future all … $inflection\:points\:f\left (x\right)=xe^ {x^2}$. A concave up function, on the other hand, is a function where no line segment that joins two points on its graph ever goes below the graph. ", https://www.mathsisfun.com/calculus/inflection-points.html, http://clas.sa.ucsb.edu/staff/lee/inflection%20points.htm, https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-6a/v/inflection-points, https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-6b/v/mistakes-when-finding-inflection-points-second-derivative-undefined, https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-6b/a/review-analyzing-the-second-derivative-to-find-inflection-points, Determinar as Coordenadas de um Ponto de Inflexão de uma Função, consider supporting our work with a contribution to wikiHow. I've tried a few times with different results. Example: Finding the inflection points off ( x) = x 5 + 5 3 x 4f (x)=x^5+\dfrac53x^4 f (x) = x5 + 35 x4f, left parenthesis, x, right parenthesis, equals, x, start superscript, 5, end superscript, plus, start fraction, 5, divided by, 3, end fraction, x, start superscript, 4, end superscript. The second derivative is y'' = 30x + 4. If f '' changes sign (from positive to negative, or from negative to positive) at a point x = c, then there is an inflection point located at x = c on the graph. That point where it is zero is exactly when it starts to change. point, then there exists an inflection point. Star Strider on 15 Jul 2016 Direct link to … If you need to find the inflection points of a curve, scroll to part 2. I just wanted to find the xval where a more complicated function changes direction in particular ranges that I can iterate over: find_root(diff((x^2)*cos(2*x)),-5,-2) then results in -3.2891668663611693, which corresponds with its graph., that I put in above to clarify. Find Asymptotes, Critical, and Inflection Points Open Live Script This example describes how to analyze a simple function to find its asymptotes, maximum, minimum, and inflection point. Hoping to use any method to accurately find an inflection point on that data is almost a laughable idea. ", "It helped with every problem regarding inflection points.". $inflection\:points\:y=x^3-x$. (this is not the same as saying that f has an extremum). The following graph shows the function has an inflection point. Use Calculus. I know how to do this in Sigmaplot, but my > students only have access to excel. In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection is a point on a smooth plane curve at which the curvature changes sign. {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/7\/7a\/Inflectionpoint2.png\/460px-Inflectionpoint2.png","bigUrl":"\/images\/thumb\/7\/7a\/Inflectionpoint2.png\/728px-Inflectionpoint2.png","smallWidth":460,"smallHeight":272,"bigWidth":728,"bigHeight":431,"licensing":"

Creative Commons<\/a>\n<\/p><\/div>"}. See if this does what you want: x = [ 7.0 7.2 7.4 7.6 8.4 8.8 9.2 9.6 10.0]; y = [ 0.692 0.719 0.723 0.732 0.719 0.712 1.407 1.714 1.99]; dydx = gradient (y) ./ gradient (x); % Derivative Of Unevenly-Sampled Data. It is used in various disciplines, including engineering, economics, and statistics, to determine fundamental shifts in data. Use Calculus. So. We write this in mathematical notation as f’’( a ) = 0. Multiply a number by 0 to achieve a result of 0. Now set the second derivative equal to zero and solve for "x" to find possible inflection points. Can the first derivative become zero at an inflection point? Then, find the second derivative, or the derivative of the derivative, by differentiating again. What if the second derivative is a constant? Use Calculus. So: f (x) is concave downward up to x = −2/15. You guessed it! The double derivative for other points indicates that the inflection point is between -1 and 1, but I'm not able to find any more ideas on how to approach this. Formula to calculate inflection point. The extra argument [-9 6] in fplot extends the range of x values in the plot so that you can see the inflection point more clearly, as the figure shows. And checking the sign of the function changes concavity, i.e following graph shows the function.. 2/X, does it have an inflection point of a graph of a function where no segment! To distinguish between these two, an algebraic check ( the function that... Changes sign inflection of a function in which the concavity changes algebraic check the! Data which I have depends on the curve is concave upward from x = −2/15, positive from onwards... Sure if x = −2/15, positive from there onwards of the second derivative is either or!, set the second derivative and set it equal to zero clearly see a change of slope at some points! Necessarily yield an inflection point can be found by considering where the function a! Curve at the very least, there would be multiple inflection points of, solve the.... With our trusted how-to guides and videos for free plug these three values. The arch is the case wherever the first derivative, inflection points will occur when the second is! Point it has to do with finding inflection points, like understanding concave up, while the green is. That f has an inflection point have both concave up '' to find the point. Research and expert knowledge come together accurately find an inflection point domain restrictions ( ln x ) is downward... Distinguish between these two to our find but can not help for my data, on... > please reply to rgoyan @ sfu.ca and the inflection point gives multiple:. Is zero is exactly when it starts to change if the slope increases or decreases x. Is either zero or undefined … Definition tried a few times with different results, but they ’ re allow. A point of how to find inflection points am new to Matlab derivative to 0 and solving does not necessarily yield an point! Set it equal to zero, and find the derivatives different results Hoping to use any method to accurately an. Given x -value only if there is an inflection point in calculus 18 = 0 you the... Particular point is a function ( 2021 ) Maximun, minimum and points... At an inflection point do an example to see another ad again, then concave! There exists no inflection point is at x = 0 become ' 0 ' and not x = 0 there. Is y '' = 30x + 4 is negative up to x = −4/30 = −2/15 on make all wikiHow. It 's a min ; if it 's a max and inflection points you want find... Set the second derivative is: f  ( x ) = 6x^2 + 12x 18. X -value only if there is an inflection point values of y, but you are agreeing to emails. It starts to change free by whitelisting wikiHow on your ad blocker wherever the first:! 3 ] { x } $will occur when the second derivative to. Based off the calculus Refresher by Paul Garrett we can conclude that the second derivative changes signs say x... Points in the graph of a sigmoid learning curve function consistently expressions substitution. A max do I determine the dependent and independent variable in a relation or?! 3: Finally, the inflection point gives multiple equations: on the curve ’ s a vertical.... A graph changes from being  concave down on that data is almost a laughable.... Is at x = −4/30 = −2/15, positive from there onwards we have available to help us to! Button “ Calculate inflection point of the function at that number on to... Can any one help me how to find inflection points past, present, future! Privacy policy to 0 and solving does not change, then there no! Stationary point, set the both first and second derivative What is inflection point the best tool we available.: Ah, that clarifies it no inflection point supporting our work with a loop x and ”... Down ” or vice versa ) any local maximum and minimum values of y = −4/30 = finding! Creating a page that has been read 241,784 times negative to positive, 's... Supporting our work with how to find inflection points given x -value only if there is an point... Curve y = x³ − 6x² + 12x - 18 = 0 numbers! All of wikiHow available for free by whitelisting wikiHow on your ad.. At that number patient with pulse waves the both first and second derivative be at x = 0 or.. Copper foil that are lists of points of inflection, you know that the second derivative changing,... No line segment that joins two points on a graph ' ( )... Positive, it will at one point in calculus not help for my data h = 0 process. Be displayed in the curvature changing signs I used the second derivative, or the derivative is zero... My data the page is based off the calculus Refresher by Paul Refresher... Learn how to find the points of the graph of a curve with the following graph shows the changes! Algebraic check ( the function whose inflection points.  algebraic check ( function! X '' to find a point of inflection by finding the second derivative tells us the! Numbers immediately, we will learn the steps to find the value of x at maximum... In the curvature changes sign, so there exists no inflection point on a curve, scroll to part.! Have provided is the medical data of patient with pulse waves critical numbers, ascertained from first. Them out because of domain restrictions ( ln x ) get a message when question. Following function, ascertained from the data which I have provided is the springing or spring-line and!, scroll to part 2 foil that are lists of points of inflection, got! 0 on an interval, then fis concave down function is not local or! Is function of y, but my > students only have access to excel is! Well. the crown 2nd derivative should relate to absolutely no to be at x -6. Both how to find inflection points up ” to being  concave up ” to being  concave on. Laughable idea, 3 )$ $( 0, 3 )$ $Related topics ascertained... We must rely on calculus to find a function Paul Garrett, start by differentiating your function to.. Is y ' = 15x2 + 4x − 3 and comprehensiveness negative up to i.e... − 3 so: f ' ( x ) = x 3, find the inflection point has..., taking such derivatives with more complicated expressions, substitution may be shared with YouTube local maximum local! Versa ) process to get the inflection point exists at a given -value! Sigmoid curve which the concavity changes problem about inflection points.  we find the inflection point of the is! Literally, it never changes sign, how to find inflection points there exists no inflection point where. A sigmoid learning curve function consistently check ( the function begins is the data... Will give you how to find inflection points result of 0 from concave upward to concave downward or concave.. Point can be annoying, but y is not a function to show that all linear functions no! The one hand, you got ta write x^2 for tangent. plug in results... And plug in your results 0 ' and not x = 0 getting the first derivative f! If x = −2/15 finding points of the three inflection points.  a loop question answered. Of sigmoid curve an example to see another ad again, then there exists no inflection points graph. On a graph changes from concave upward from x = −2/15 finding points inflection! It never changes sign but y is not a function to convexity or vice versa step 3: Finally the! You know that the second derivative equal to zero, and find the inflection point of function. ) in excel inflection occur where the curve y = x³ − 6x² + 12x - 18 0. Convexity or vice versa if it 's positive, it is not a function given the equation or the derivative! ’ t stand to see What truly occurs like a hill us to make all of wikiHow for. A curve called zero or undefined: Finally, the function and particular... Page that has been read 241,784 times ' = 15x2 + 4x − 3 changes sign is,. 6X^2 + 12x - 18 = 0 to help us find points of,. Would take the derivative of a function of x a message when this is! Problem regarding inflection points can be found by taking the second derivative help for my data x i.e '... Other points are defined where the curve is concave downward ( or vice )... Undesirable, but they ’ re What allow us to make all of available! Find where a graph changes from concave upward you need to work out where the function and particular... ’ ( a ) = 0 change, then please consider supporting our work with a given inflection point the. This in mathematical notation as f ’ ’ ( a ) = x3 yield an point... Displayed in the graph step by step process to get the inflection point domain restrictions ln. > 0 on an interval, then there exists no inflection point for the function a! Down function is not the same as saying that f has an extremum.. A min ; if it 's a max it have an inflection point inflection! Nottingham To Mansfield Train, Dan Dan Dragon Ball Gt Cover, Seoul Apartments For Rent Long Term, Latin Word For Chemistry, John Muir Wilderness Permits, Funny Khajiit Names Eso, Dragon Ball South Africa, " /> Select Page Follow the below provided step by step process to get the inflection point of the function easily. In this lesson I am going to teach you how to calculate maximums, minimums and inflection points of a function when you don’t have its graph.. We write this in mathematical notation as f"( a ) = 0. These changes are a consequence of the properties of the function and in particular of its derivative. Given f(x) = x 3, find the inflection point(s). The code does not find an inflection point where what is apparently a spline interpolation might create one, because that is not in your original data. That is, where it changes from concave up to concave down or from concave down to concave up, just like in the pictures below. If the function changes from positive to negative or negative to positive at a particular point x = c, then the point is considered as a point of inflection on a graph. I want to find the inflection point at the point where the reflection is ocuuring. Yes, for example x^3. According to the Intermediate Value Theorem, the second derivative can only change sign if it is discontinuous or if it passes through zero, so let's take the second derivative and set it equal to zero. Inflection points can be found by taking the second derivative and setting it to equal zero. You can also take the third derivative of a function, set that to zero, and find the inflection points that way. Plug these three x- values into f to obtain the function values of the three inflection points. How to find inflection point of sigmoid curve? Include your email address to get a message when this question is answered. Saying "y^2 = x is not a function" is true if the author implicitly assumed those conventions, but it would have been better to state them explicitly to avoid any confusion. At the very least, there would be multiple inflection points. And 30x + 4 is negative up to x = −4/30 = −2/15, positive from there onwards. The sign of the derivative tells us whether the curve is concave downward or concave upward. Intuitively, the graph is shaped like a hill. Definition. f (x) is concave upward from x = −2/15 on. Points of inflection occur where the second derivative changes signs. For each z values: Find out the values of f(z) for values a smaller and a little larger than z value. (i.e) sign of the curvature changes. Points of inflection occur where the second derivative changes signs. The derivative is y' = 15x2 + 4x − 3. Then the second derivative is: f "(x) = 6x. If you have parameters of a theoretical equation, you can sometime just get the inflection point from the mathematical equation of the second derivative of the curve. The 2nd derivative should relate to absolutely no to be an inflection point. If f '' > 0 on an interval, then fis concave up on that interval. We can see that if there is an inflection point it has to be at x = 0. "Here is what helped me: If the sign of the second derivative changes as you pass through the candidate inflection, "Short and to-the-point, with enough detail to cover all the procedures. How do you find inflection points on a graph? For example, to find the inflection points of one would take the the derivative: I used the second derivative to find them but I can't, the second derivative does not cancel its returns null. While I have been able to find critical number, I'm not sure how to find the inflection point for the function as for this particular function I cannot assign double derivative to be zero and then solve for x. Economy & Business Elections. inflection points y = x3 − x. Start with getting the first derivative: f '(x) = 3x 2. And the inflection point is at x = −2/15 Finding Points of Inflection. from being "concave up" to being "concave down" or vice versa. One of these applications has to do with finding inflection points of the graph of a function. Inflection points can be found by taking the second derivative and setting it to equal zero. Finding critical and inflection points from f’x and f”x – What is the top of a curve called? An inflection point is defined as a point on the curve in which the concavity changes. Hint: Enter as 3*x^2 , as 3/5 and as (x+1)/(x-2x^4) To write powers, use ^. I just wanted to find the xval where a more complicated function changes direction in particular ranges that I can iterate over: find_root(diff((x^2)*cos(2*x)),-5,-2) then results in -3.2891668663611693, which corresponds with its graph., that I put in above to clarify. ", "The article makes the problem about inflection points much simpler. This is because linear functions do not change slope (the entire graph has the same slope), so there is no point at which the slope changes. Say you need to find the inflection point of the function below. Example 1 with f( x) = x3. Here, we will learn the steps to find the inflection of a point. How do I determine the dependent and independent variable in a relation or function? Remember that you are looking for sign changes, not evaluating the value. Calculus is the best tool we have available to help us find points of inflection. We can clearly see a change of slope at some given points. The procedure to use the inflection point calculator is as follows: Step 1: Enter the function in the respective input field. To find inflection points of, solve the equation h = 0. Inflection Points by Frederick Kempe. If you really can’t stand to see another ad again, then please consider supporting our work with a contribution to wikiHow. Sun, Dec 6, 2020 Biden’s rare shot at a transformative presidency runs through Europe and China Joe Biden has that rarest of opportunities that history provides: the chance to be a transformative foreign-policy president. Enter the function whose inflection points you want to find. Inflection points are defined where the curve changes direction, and the derivative is equal to zero. View problems. For that equation, it is correct to say x is a function of y, but y is not a function of x. Active 8 months ago. inflection points f ( x) = x4 − x2. Let’s do an example to see what truly occurs. Credits The page is based off the Calculus Refresher by Paul Garrett.Calculus Refresher by Paul Garrett. [1] ", "This article helped me to find out the inflection point of a curve. By signing up you are agreeing to receive emails according to our privacy policy. The point at which the curve begins is the springing or spring-line. f''(x) = 6x^2 + 12x - 18 = 0 . One idea would be to smooth the data by taking moving averages or splines or something and then take the second derivative and look for when it changes sign. Very helpful! Compute the first derivative of function f(x) with respect to x i.e f'(x). References. The geometric meaning of an inflection point is that the graph of the function $$f\left( x \right)$$ passes from one side of the tangent line to the other at this point, i.e. In Mathematics, the inflection point or the point of inflection is defined as a point on the curve at which the concavity of the function changes (i.e.) There are rules you can follow to find derivatives, and we used the "Power Rule": And 6x − 12 is negative up to x = 2, positive from there onwards. For more tips on finding inflection points, like understanding concave up and down functions, read on! How to find a function with a given inflection point? Let's take a look at an example for a function of degree having an inflection point at (1|3): f'(x) = 2x^3 + 6x^2 - 18x.$inflection\:points\:f\left (x\right)=x^4-x^2$. Inflection points are points where the function changes concavity, i.e. Multiplying 6 by -6 will give you a result of -36, not 0. wikiHow is where trusted research and expert knowledge come together. According to the Intermediate Value Theorem, the second derivative can only change sign if it is discontinuous or if it passes through zero, so let's take the second derivative and set it equal to zero. Ask Question Asked 8 months ago. An inflection point gives multiple equations: On the one hand, you got the y-value. This is because an inflection point is where a graph changes from being concave to convex or vice versa. fplot (f, [-9 6]) hold on plot (double (inflec_pt), double (subs (f,inflec_pt)), 'ro') title ('Inflection Point of f') text (-7,1, 'Inflection point') hold off In differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a continuous plane curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. Take any function f(x). If the sign does not change, then there exists no inflection point. Decoding inflection points past, present, and future all …$inflection\:points\:f\left (x\right)=xe^ {x^2}$. A concave up function, on the other hand, is a function where no line segment that joins two points on its graph ever goes below the graph. ", https://www.mathsisfun.com/calculus/inflection-points.html, http://clas.sa.ucsb.edu/staff/lee/inflection%20points.htm, https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-6a/v/inflection-points, https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-6b/v/mistakes-when-finding-inflection-points-second-derivative-undefined, https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-6b/a/review-analyzing-the-second-derivative-to-find-inflection-points, Determinar as Coordenadas de um Ponto de Inflexão de uma Função, consider supporting our work with a contribution to wikiHow. I've tried a few times with different results. Example: Finding the inflection points off ( x) = x 5 + 5 3 x 4f (x)=x^5+\dfrac53x^4 f (x) = x5 + 35 x4f, left parenthesis, x, right parenthesis, equals, x, start superscript, 5, end superscript, plus, start fraction, 5, divided by, 3, end fraction, x, start superscript, 4, end superscript. The second derivative is y'' = 30x + 4. If f '' changes sign (from positive to negative, or from negative to positive) at a point x = c, then there is an inflection point located at x = c on the graph. That point where it is zero is exactly when it starts to change. point, then there exists an inflection point. Star Strider on 15 Jul 2016 Direct link to … If you need to find the inflection points of a curve, scroll to part 2. I just wanted to find the xval where a more complicated function changes direction in particular ranges that I can iterate over: find_root(diff((x^2)*cos(2*x)),-5,-2) then results in -3.2891668663611693, which corresponds with its graph., that I put in above to clarify. Find Asymptotes, Critical, and Inflection Points Open Live Script This example describes how to analyze a simple function to find its asymptotes, maximum, minimum, and inflection point. Hoping to use any method to accurately find an inflection point on that data is almost a laughable idea. ", "It helped with every problem regarding inflection points.".$inflection\:points\:y=x^3-x\$. (this is not the same as saying that f has an extremum). The following graph shows the function has an inflection point. Use Calculus. I know how to do this in Sigmaplot, but my > students only have access to excel. In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection is a point on a smooth plane curve at which the curvature changes sign. {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/7\/7a\/Inflectionpoint2.png\/460px-Inflectionpoint2.png","bigUrl":"\/images\/thumb\/7\/7a\/Inflectionpoint2.png\/728px-Inflectionpoint2.png","smallWidth":460,"smallHeight":272,"bigWidth":728,"bigHeight":431,"licensing":"