Determine whether the Mean Value Theorem can be applied to the function on the closed interval .If the Mean Value Theorem can be applied,find all numbers c in the open interval such that .
A)MVT applies;
B)MVT applies;
C)MVT applies;
D)MVT applies;
E)MVT applies;

Question

Determine whether the Mean Value Theorem can be applied to the function  on the closed interval  .If the Mean Value Theorem can be applied,find all numbers c in the open interval  such that  . ​
A)MVT applies; 
B)MVT applies; 
C)MVT applies; 
D)MVT applies; 
E)MVT applies;

Quizplus · Accepted Answer

The function is continuous on the closed interval and differentiable on the open interval. Therefore, the Mean Value Theorem can be applied. By the Mean Value Theorem, there exists at least one number c in the open interval (0,1) such that f'(c)=(f(1)-f(0))/(1-0). In this case, we have f(x)=x^3-2x+1, f(1)=0 and f(0)=1, so (f(1)-f(0))/(1-0)=-1. We need to find a number c in the open interval (0,1) such that f'(c)=-1. We have f'(x)=3x^2-2, so we need to solve the equation 3c^2-2=-1, which gives c=1/sqrt(3). Therefore, the answer is A.

Determine Whether the Mean Value Theorem Can Be Applied to the Function