Question

Determine whether the Mean Value Theorem can be applied to ?(?) = ?^3 − 3? + 2 on the closed interval [−2,2].

If the Mean Value Theorem can be applied, find all values of ?? in the open interval (−2,2) such that

?′ (?) = ?(2) − ?(−2) /2 − (−2)

If the Mean Value Theorem cannot be applied, explain why not.

Answer 1

Solution :- we have  ?(?) = ?^3 − 3? + 2

The main conditions of the Mean Value Theorem are:

• f(x) must be continuous on [a, b]
• f(x) must be differentiable on (a, b).
• There is some point c on (a, b) that satisfies.

We can assume that conditions 1 and 2 are fulfilled since you have a polynomial, and polynomials are differentiable (and continuous) everywhere . Also f'(c) = [f(b) - f(a)]/(b - a)

so ?′ (?) = {?(2) − ?(−2)}  /[ 2 − (−2)]

then 3 c^2 - 3 = (4 - 0)/(4) = 1

thus we get

3x^2 = 4

c = sqrt(4/3) and c = + 2/ sqrt(3) and c = -2/ sqrt(3)