Question #293161

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


1
Expert's answer
2022-02-03T11:46:53-0500

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

f(x)=x55x4+5x31,f(x)=5x420x3+15x2, and f(0)=0f(0) is a stationary value but f(0) is not an \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,f0.f''=0, f'''\ne 0.


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