Question

What are the local extrema, if any, of `f(x)= 120x^5 - 200x^3`?

Answer 1

Local maximum of `80` (at `x=-1`) and local minimum of `-80` (at `x=1`.

Explanation:

`f(x)= 120x^5 - 200x^3`

`f'(x)= 600x^4 - 600x^2 = 600x^2(x^2 - 1)`

Critical numbers are: `-1`, `0`, and `1`

The sign of `f'` changes from + to - as we pass `x=-1`, so `f(-1) = 80` is a local maximum.

(Since `f` is odd, we can immediately conclude that `f(1)=-80` is a relative minimum and `f(0)` is not a local extremum.)

The sign of `f'` does not change as we pass `x= 0`, so `f(0)` is not a local extremum.

The sign of `f'` changes from - to + as we pass `x=1`, so `f(1) = -80` is a local minimum.