Suppose a function f(x) is differentiable and continuous on the interval [a,b]. If there exists a number c in the interval [a,b] such that f '(c) = 0, then the Mean Value Theorem applies.
For example.. suppose f(x) = x^2 + 3x. We want to test the Mean Value Theorem over the interval [-2, 1].
f'(x) = 2x + 3
Now set f'(x) = 0 and solve for x.
2x + 3 = 0, therefore x = -3/2. Since -3/2 falls in the interval, the Mean Value Theorem applies.
An example where it doesn't apply. Suppose the interval is [-5,5]
f(x) = 2x..
f'(x) = 2
there is no value c where the derivative is 0 since the derivative everywhere along the function is 2. Therefore, the Mean Value Theorem does not apply.