Question

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers  that satisfy the conclusion of the Mean Value Theorem.

f (x) = ∛x,       [0, 1]

Answer 1

Step 1:

The function is , .

Mean value theorem :

Let f be a function that satisfies the following three hypotheses :

1. f is continuous on

2. f is differentiable on

Then there is a number c in such that

.

Step 2:

The function is .

The function is  continuous on the interval .

Differentiate with respect to .

The function is differentiable on the interval .

Then .

Step 3:

From the mean value theorem :

.

Substitute in .

Rationalize the denominator with .

Solution :

.