Question

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.)

f(x) = x³ - 4x² - 16x + 9, [4, 4]
Find c=

Answer 1

There seems to be an error in the interval [4, 4] as it represents a single point, not an interval. To apply Rolle's theorem, we need a proper interval. Without a valid interval, we cannot determine the number c that satisfies the theorem.

Step-by-step explanation:

To apply Rolle's theorem, a set of conditions need to be fulfilled: the function must be continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) must equal f(b).

The question presents the function f(x) = x³ - 4x² - 16x + 9 and asks to verify these conditions on the interval [4, 4]. However, this seems to be a typo because the interval [4, 4] suggests a single point rather than an interval. Even if we correct the interval, we can't proceed with Rolle's theorem unless the interval represents more than a single point.

If the interval was indeed a proper interval and the conditions were satisfied, we would find the value c by taking the derivative f'(x) of our function and setting it equal to zero to solve for c in the interval.

Without a valid interval, we cannot find such a number c that satisfies the conclusion of Rolle's Theorem. An accurate interval is necessary to tackle this problem.