Question

Why is a point, b, an extremum of a function if `f'(b)=0`?

Answer 1

A point at which the derivative is `0` is not always the location of an extremum.

Explanation:

`f(x)=(x-1)^3 = x^3-3x^2+3x-1`

has `f'(x) = 3(x-1)^2 = 3x^2-6x+3`,

so that `f'(1)=0`.

But `f(1)` is not an extremum.

It is also NOT true that every extremum occurs where `f'(x)=0`
For example, both `f(x) = absx` and `g(x)=root3(x^2)` have minima at `x=0`, where their derivatives do not exist.

It IS true that if `f(c)` is a local extremum, then either `f'(c)=0` or `f'(c)` does not exist.