# Is the inflection point being always the mid-point of two extremes?

Is it necessarily that an inflection point always the mid-point between two critical points for any twice differentiable function?

I have seen this in YouTube video related to concavity, I fully understand what is inflection point, it is the point where concavity changes (concave up then concave down) or (concave down then concave up). What is the truth of the statement above? Is there any proof or disproof?

Any help would be appreciated. THANKS!

• Applies to functions that are cubic in nature, since their second derivative is linear. – Andrew Chin Mar 19 '20 at 22:28
• Try $f(x) = x^3$. It has one critical point, which is also an inflection point. – Matthew Leingang Mar 19 '20 at 22:28

Let $$f(x) = x^3(x-1)$$, which is easily seen to be twice differentiable. The critical points are $$\{0,\frac{3}{4}\}$$ and the inflection points are at $$\{0,\frac{1}{2}\}$$.