The inflection point is where the 2. derivative is zero.

f = AX^B + CX^D

f' = B*A*X^(B-1) + D*C*X^(D-1)

f'' = (B-1)*B*A*X^(B-2) + (D-1)*D*C*X^(D-2)

The 2. derivative is zero for x0 = (-(D-1)*D*C / ((B-1)*B*A)) ^ (1/(B-D)). Evaluating this with your values for A,B,C,D gives x0 = 18.81 and f(x0) = 0.98.

Btw: The function has extreme values where the first derivative is zero. This is at at xe = (-(D*C)/(B*A))^(1/(B-D)), what evaluates to 12, with f(xe) = 0.92. Since the function at the extremum is lower than the function at the inflection point, it must be a minimum. This can be checked formally by looking at the values of the 2nd derivative at X = xe, which exists and > 0 (f''(12) = 0.0065), indicating that it really is a minimum.

Hello Dears,

I don't think so , when you make function plot, we should find such graph , so the point found by dear professor Jochen Wilhelm I think is not true.

Dear Chellai Fatih ,

you have choosen a subptimal scale. The function goes steeply towards infinity for x close to 0. Have a closer look at x in (8, 30) to see the point. I attached a figure showing this range.

yes dear professor Jochen Wilhelm , you are totaly right, I commit a calculus mistake.

Best wishes dear.

Thank you for the prompt answer Jochen Wilhelm . This has helped a lot. Is there any literature you can recommend on this topic? It is interesting that you mention that 12 is the minimum as it appears to be the inflection point when visually assessing the curve. Please see the red curve in the attached file.

Caroline Candebat ,

the "topic" is curve sketching. It's 10th grade math stuff...

https://en.wikipedia.org/wiki/Curve_sketching

I don't get your note that the red curve shows an inflection point at x = 12. At this value, the red curve has a minimum. The inflection point seems to be at around 19. To find the inflection point, imagine you drife along the red curve from left to right with a car, say. In the beginning, you would need to steer left (you are drifing a left curve). After passing x=12 you will slowly turn the steeing wheel rightwards. When, at about x=19, you reach the inflection point, the steeing wheel is just in the center position, and as you drive on you start going into a right curve.

It seems that the red curve is a fit (an estimate) based on some noisy data. Note that the estimate of the inflection point (as well as the minimum) is accociated with considerable uncertainty. It may be helpful to provide a standard error or a confidence interval for these estimates.

Jochen Wilhelm I doubt that differential calculus is taught in the 10th grade at least in the US. For those that can't sleep nights without learning it let me suggest the small book that got me through my first semester. The theoretical treatment of limits is very weak but it suffices for understanding and as the author says "the rest will follow". This book changed my life. I hope that you enjoy it too..

https://www.google.com/search?q=Calculus+made+Easy+Pdf&rlz=1C1CHBF_enUS874US874&oq=Calculus+made+Easy+Pdf&aqs=chrome..69i57.15501j0j1&sourceid=chrome&ie=UTF-8

Best wishes for the new year, D. Booth

David Eugene Booth You are right, 10th grade was too early. I looked it up: it's actually 11th grade in Germany. However,at some time during school, everyone doing a bachelor, master or PhD should have heard it (and ideally be familiar with it -- to some exted ;) ). Not sure about that but I think it should also be taught again in the first semesters of university when studying empirical sciences (or econometrics and maths, for sure).

Your book looks like a valuable aid to learn calculus. Thanks for the link!

David Eugene Booth Thank you for sharing this book with me and the researchgate community.

For Inflection point or maxima/minima or extreme points or boundary points or equilibrium points :

1) First derivative of the given function.

2) Apply initial conditions.

3) Get the points.

4) Sketch it or code it and visualize representative graph for the given equation.