answer is only . It’s very important to check that the answers you get
for c fall in the given interval when doing Mean Value Theorem problems.
5.
Rolle’s Theorem says that if 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 exists at least one number c
on the interval (a, b) such that f′(c) = 0. Here the function is f(x) = x^3 − x and the interval is
[−1, 1]. First, we check if the function is equal to zero at both of the endpoints: f(−1) = (−1)^3 −
(−1) = 0 and f(1) = (1)^3 − (1) = 0. Next, we take the derivative to find f′(c): f′(x) = 3x^2 − 1, so f
′(c) = 3c^2 − 1. Now, we can solve for c: 3c^2 − 1 = 0 and c = . Note that are both in
the interval (−1, 1), just as we expected.
- $15 billion
We simply take the derivative: = 3x^2 − 96x + 720. Now we set the derivative equal to
zero: 3x^2 − 96x + 720 = 0. The solutions to this are x = 12 and x = 20. Note, however, that the
function is bounded by x = 0 and x = 40.
