Question

Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply.)

f(x) = −x2 + 2x, [0, 2]
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).

Answer 1

A) Yes, Rolle's Theorem can be applied!

Explanation:

Rolle's theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b.

Here, for our continuous function
`f(x)=-x^2+2x` over the closed interval
`[0,2]`, we can tell that the function is clearly differentiable over the interval
`(0,2)` as
`f'(x)=-2x+2`, so we'll need to check if
`f(0)=f(2)`:

`f(0)=-0^2+2(0)=0\\f(2)=-(2)^2+2(2)=-4+4=0`

Next, we'll need to check if f'(x) = 0 for some x within the closed interval:

`f'(x)=-2x+2=0\\-2x+2=0\\-2x=-2\\x=1`

As x=1 is contained in [0,2] and the previous conditions were met, Rolle's Theorem can be applied!