Question

Does Rolle's theorem apply to the function f(x)=x³-9x on the interval [-3,0]? If so, find all numbers c on the interval that satisfy the theorem.

Answer 1

Rolle's Theorem applies to the function f(x) = x³ - 9x on the interval [-3,0] because the function is continuous and differentiable on the interval and the endpoints have equal values. By finding the zeros of the derivative, number c within the interval satisfying the theorem is c = -√3.

Step-by-step explanation:

The question asks whether Rolle's Theorem can be applied to the function f(x) = x³ - 9x on the interval [-3,0] and if so, to find all numbers c that satisfy the theorem within this interval.

To apply Rolle's Theorem, three conditions must be met:

• The function f(x) must be continuous on the closed interval [a, b].
• The function must be differentiable on the open interval (a, b).
• The function must have equal values at the endpoints of the interval, i.e., f(a) = f(b).

For the given function f(x) = x³ - 9x:

1. It is a polynomial, so it is continuous and differentiable everywhere, which means it is continuous on [-3,0] and differentiable on (-3,0).
2. Calculating the function value at the endpoints gives us f(-3) = (-3)³ - 9(-3) = -27 + 27 = 0 and f(0) = 0³ - 9(0) = 0, so f(-3) = f(0).

Since all three conditions are met, Rolle's Theorem applies, and there exists at least one number c in (-3,0) such that f'(c) = 0. To find c, we differentiate the function:

f'(x) = 3x² - 9.

Setting the derivative equal to zero:

3x² - 9 = 0

x² = 3

x = ±√3

Since only the negative square root is within the interval, c = -√3.