Then
f′(c) = There is no value of c that will satisfy this equation! We expected this. Why? Because f(x) is not
continuous at x = 0, which is in the interval. Suppose the interval had been [1, 3], eliminating the
discontinuity. The result would have been
c = − is not in the interval, but c = is. The answer is c = .
Example 4: Consider the function f(x) = x^2 − x − 12 on the interval [−3, 4].
Follow the MVTD.
f′(c) = = 0 and f′(c) = 2c − 1 = 0, so c = Note: This is a great way to self-check your work. Always look
at whether your answer makes sense.In this last example, you discovered where the derivative of the equation equaled zero. This is going to be
the single most common problem you’ll encounter in differential calculus. So now, we’ve got an important
tip for you.
When you don’t know what to do, take the derivative of the equation and set it equal to zero!!!Remember this advice for the rest of AP Calculus.
ROLLE’S THEOREM
Now let’s learn Rolle’s Theorem, which is a special case of the MVTD.