Help with derivatives?

Question

I need some help with this problem...

Let f(x) = |x^3 - 9x|

In words, f of x equals the absolute value of x cubed minus nine x.

a) Does f ' (0) exist?

b) Does f ' (3) exist?

c) Does f ' (-3) exist?

d) Determine all the extrema for f.

Thanks in advance.

Answer 1

The way to construct the solution is to make f(x)=f(x)sgnf(x), where for real values of f(x) f(x)=-f(x) f(x)<0 and f(x)=f(x) f(x) >0

Now solve for f(x)=0, x=0, 3, -3
for x <-3 the signed function goes negative, which means that the absolute value will undergo a sign change and have the form 9x-x^3 as x decreases.
from -3 to 0 it is positive, and from 0 to 3 it is negative again, so it will be 9x-x^3 from three greater it is positive again.

Does the f' exist at the v points where the sgn function causes a extreme change in direction, no. The derivative is called "undefined".

The max value for f is infinity, and the min is zero.