Question

Using Rolle's Theorem In Exercises 9-22, determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f'(c)0. If Rolles Theorem cannot be applied, explain why not.

f(x) = -x² + 3x. [0, 3]

Answer 1

Rolle's Theorem can be applied to f(x) = -x² + 3x on the interval [0, 3] since the function is continuous, differentiable, and has equal values at the endpoints. The value of c where the derivative is zero within the interval is 1.5.

Step-by-step explanation:

The student asks if Rolle's Theorem can be applied to the function f(x) = -x² + 3x on the closed interval [0, 3].

To apply Rolle's Theorem, three conditions must be met: the function must be continuous on the closed interval [a, b], differentiable on the open interval (a, b), and the function's values at the endpoints of the interval must be equal.

For the given function, f(x) is continuous and differentiable everywhere because it is a polynomial.

At the endpoints, f(0) = -(0)² + 3(0) = 0 and f(3) = -(3)² + 3(3) = 0. Since the function satisfies all the conditions, Rolle's Theorem can be applied.

To find the value of c in the interval (0, 3) where f'(c) = 0, we first find the derivative of the function, f'(x) = -2x + 3. Setting the derivative equal to zero gives -2c + 3 = 0, which results in c = 1.5.

Therefore, the value of c where f'(c) = 0 is 1.5.