Can Rolle's theorem be applied or no?

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 on 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

Rolle’s Theorem states that if f is a continuous function on the closed interval [a,b], differentiable in the open interval (a,b), and f(a)=f(b) then there exists at least one number c in (a,b) such that the f’(c) = 0.

Rolle’s Theorem is a simple three-step process:

1. Check to make sure the function is continuous and differentiable on the closed interval.
2. Plug in both endpoints into the function to check they yield the same y-value.
3. If yes, to both steps above, then this means we are guaranteed at least one point within the interval where the first derivative equals zero.

Step 1.

Since f(x) = −x² + 6x is a polynomial function, then f(x) is continuous and differentiable.

Step 2.

plug in 0 and 6 to the function to check if they have the same value

f(0) = 0

f(6) = -6²+6(6) = 0

Both Steps 1 and 2 are valid, which means Rolle's theorem applies

f'(x) = -2x+6

for values c

f'(c) = -2c+6 = 0

Thus c = 3