Question

Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = −x2 + 6x, [0, 6] Yes, Rolle's Theorem can be applied. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). No, because f(a) ≠ f(b). If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f '(c) = 0. (Enter your answers as a comma-separated list. If Rolle's Theorem cannot be applied, enter NA.) c =

Answer 1

Answer:

• The theorem applies on the given function  $f(x) = -x^2 + 6x$.
• c=3

Step-by-step explanation:

We are given that: $f(x) = -x^2 + 6x$ on the closed interval: [0,6]

Rolle's Theorem holds if the following conditions are satisfied:

• The function must be continuous in [a,b]
• The function must be differentiable in (a,b)
• If f(a) = f(b), and there exist c in (a,b) such that $f'(c)=0$

Continuity of function

Since the function $f(x) = -x^2 + 6x$ is a continuous function, it is continuous everywhere. Therefore, f(x) is continuous in [0,6]

Differentiability of function

The polynomial $f(x) = -x^2 + 6x$ is differentiable on [0,6] since $f'(x) = -2x + 6$

Next, we evaluate f(0) and f(6)

$f(0) = -(0)^2 + 6(0)=0\\f(6) = -6^2 + 6(6)=0$

Next, we determine if c is in (a,b)

$f'(x) = -2x + 6$

$f'(c) = -2c + 6=0\\-2c+6=0\\-2c=-6\\c=3$

Therefore, c=3 is in (0,6)

• Thus, the theorem applies in the given function f(x).
• c=3