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

1 Answer
May 1, 2017

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.