Question

A differentiable function `f` has only one critical number: `x=-3`. Identify the relative extrema of `f` at (-3, f(-3)) if `f'(-4)=(1/2)` and `f'(-2)=-1`?

Answer 1

`f(-3)` is a relative maximum.

Explanation:

`f'(-3)` is either undefined or `0`.

Since `f` is differentiable, `f'` is never undefined. That, together with "`-3` is a critical number", implies that `f'(-3) = 0`.

Derivative have the intermediate value property.

Therefore, since `f'(-4) > 0` , there is nowhere in `[-4,0)` where `f'` is `0`, we can conclude that `f'(x) > 0` for `x in`[-4,-3)`so`f`is increasing on`[-4,-3)#.

Similarly, since `f'(-2) < 0` , we can conclude that `f'(x) < 0` for `x in`(-3,-2]`so`f`is decreasing on`(-3,-2]#.

Therefore `f(-3)` is a relative maximum.