# How do you determine whether the function satisfies the hypotheses of the Mean Value Theorem for #f(x)=x^(1/3)# on the interval [-5,4]?

##### 1 Answer

It doesn't. But it does satisfy the conclusion. See below.

#### Explanation:

There are two hypotheses for MVT.

The function must be continuous on

The function must be differentiable on

The first is true (satisfied) because

This function fails to be differentiable at

This function does not satisfy the hypotheses of the Mean Value Theorem on this interval.

**Bonus material**

This function DOES satisfy the conclusion of the MVT on this interval. We cannot use the Mean Value Theorem to conclude that there is a

We can, however solve

The algebra is be tedious to type, but the graph makes this plausible.

It shows

graph{(y-x^(1/3)) (y-4^{1/3)-((4^(1/3)+5^(1/3))/9)(x-4))= 0 [-6.06, 5.04, -2.864, 2.685]}