Determine if the Mean Value Theorem applies to the function on the given interval. If the either applies, find all values of c that are guaranteed by the theorem
f(x)=x^3-x^2-2x on [-1,1]

Question

Determine if the Mean Value Theorem applies to the function on the given interval. If the either applies, find all values of c that are guaranteed by the theorem 
f(x)=x^3-x^2-2x on [-1,1]

science0229 · Answer

First find f'(x), which is 3x^2-2x-2.
Now, you have to find all values of c that satistify$$\frac{ f(1)-f(-1) }{ 1-(-1) }=\frac{ -2-0 }{ 2 }=-1=f'(c)=3c^2-2c-2$$
Solve for c (-1<c<1).

mathmale · Answer

We need to ask ourselves, first, whether the function f(x) = 3x^2-2x-2 is continuous and differentiable on the given interval; those are the conditions on which the Mean Value Theorem is based.