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

1 Answer
May 1, 2017

Answer:

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.