Question

Determine whether the mean value Theorem can be applied to f on the closed interval [a, b]. If the mean value Theorem can be applied, find all values of c in the open interval (a, b) such that f ' (c) = f (b) - f (a)/(b - a). If the mean value Theorem cannot be applied, explain why not.

f (x) = x^3 + 2x,              [-1, 1]

Answer 1

Step 1:

The function is .

Mean value theorem :

Let be a function that satisfies the following hypotheses :

1. is continuous on .

2. is differentiable on .

Then there is a number  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 .

The mean value theorem satisfies the hypothesis.

Then

Step 3:

From the mean value theorem :

.

Substitute in .

Substitute in .

Solution :

The function satisfies the mean value theorem on the interval .

.

and .