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]

There is a value of in (-1, 1), .

Step-by-step explanation:

Let for , we need to prove that is continuous and differentiable to apply the Mean Value Theorem. Given that is a polynomical function, its domain comprises all real numbers and therefore, function is continuous.

If is differentiable, then exists for all value of . By definition of derivative, we obtain the following expression:

(Eq. 2)

The derivative of a cubic function is quadratic function, which is also a polynomic function. Hence, the function is differentiable at the given interval.

According to the Mean Value Theorem, the following relationship is fulfilled:

(Eq. 3)

If we know that , and , then we expand the definition as follows:

There is a value of in the interval (-1, 1), .

he spend 5.1 minutes chatting with his friends.

step-by-step explanation:

2/5+1/2=9/10

6/1-9/10=5.1

3, 12, 48, 384, 1536, 6144, 24576, 98304

step-by-step explanation:

just keep timing it by 4.