Question

How do you find the local extremas for `f(x)= (x-3)^3`?

Answer 1

No local extrema.

Explanation:

Take the derivative of `f(x)`. Use chain rule (although the derivative of `x-3` is just `1` so it doesn't really matter.:

`f'(x)=3(x-3)^2`

Find when `f'(x)=0`

`0=3(x-3)^2`

This happens when `x=3`.

Now check both sides of the zero.

However, check when you check both sides of `f'(x)` they're both positive. So there's no local extremas.

You can check this with the graph.

graph{(x-3)^3 [-10, 10, -5, 5]}

As you can see, there's never a change in slope based on this graph. There are however, absolute maximums and minimums at `x=oo` and `x=-oo` respectively.