If y = f(x) is continuous on the interval [a, b], and is differentiable everywhere on the interval (a, b),
and if f(a) = f(b) = 0, then there is at least one number c between a and b such that f ́(c) = 0.Graphically, this means that a continuous, differentiable curve has a horizontal tangent between any two
points where it crosses the x-axis.
Example 4 was an example of Rolle’s Theorem, but let’s do another.
Example 5: Consider the function f (x) = − 6x on the interval [0, 12].
First, show that
f(0) = −6(0) = 0 and f(12) = − 6(12)= 0Then find
f′(x) = x − 6, so f′(c) = c − 6If you set this equal to zero (remember what we told you!), you get c = 6. This value of c falls in the
interval, so the theorem holds for this example.
As with the MVTD, you’ll run into problems with the theorem when the function is not continuous and
differentiable over the interval. This is where you need to look out for a trap set by ETS. Otherwise, just
follow what we did here and you won’t have any trouble with either Rolle’s Theorem or the MVTD. Try
these example problems, and cover the responses until you check your work.
PROBLEM 1. Find the values of c that satisfy the MVTD for f(x) = x^2 + 2x − 1 on the interval [0, 1].