Theorem 3.4 The Mean Value Theorem
If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that
.
Theorem 3.3 Rolle's Theorem
Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f(a)=f(b) then there is at least one number c in (a,b) such that f '(c)=0.
First Rolle's Theorem Check
in the interval
c = 0
F is also continuous and differentiable on the closed interval [-2,2]. Therefore you can apply Rolle's Theorem.
Lets Check Mean Value Theorem
f is continuous and differentiable on the closed interval [-2, 2].
Because f satisfies the condtions of the Mean Value Theorem, there exists a least one number c in (-2, 2) such that f '(c)=0. Solving the equation f '(x)=0 yields f '(x)=2x=0 which implies that x=0. So in the interval (-2, 2) you can conclude that c=0.