# Derivative at the point of inflection

Is it always true that when a function changes concavity from concave-up to concave-down, its derivative at the point of inflection is undefined? And in the reverse order the derivative is Zero?

• Not true. Take $f(x)=x^3$ and look at the inflection point at $x=0$ and then look at $-f(x)$ – marwalix Jun 9 '15 at 18:05

Not at all. The derivative is well defined.

At the point of inflection derivative value or slope is a maximum or minimum. The derivative is constant. At this point the line is curving neither to left nor right but is headed straight without any turning.

For curve $y = \tan x$ or $y= \sin x$ the slope or derivative is $1$ at the origin, $y^{''} (x)$ vanishes. But please look at $y= x^3$ only after sometime to avoid the present confusion.