Question

Prove that every (relative) extreme value of a function is a stationary value but not every stationary value needs to be an extreme value.

Answer 1

Solution:

Every (relative) extreme value of a function is a stationary value but not every stationary value needs to be an extreme value.

It is true and can be verified through following example.

Example-Let

$\begin{aligned} \begin{aligned} f(x) =x^{5}-5 x^{4}+5 x^{3}-1, \\ f^{\prime}(x) =5 x^{4}-20 x^{3}+15 x^{2}, \\ \text { and } f^{\prime}(0) &=0 \\ \therefore f(0) \text { is a stationary value but } f(0) \text { is not an } \end{aligned} \end{aligned}$

extreme value because $f''=0, f'''\ne 0.$