Determine whether Rolle's Theorem can be applied to the function on the given interval; if so, find the value (s) of c guaranteed by the theorem. (Enter your answers as a comma-separated list. If Rolle's Theorem does not apply, enter DNE.) f (x) = x (5 - x) on [0, 5]

Step-by-step explanation:

1) Check if the function is differentiable on that interval. In this case, yes, because all polynomials are differentiable.

2) plug in the bounds of the interval to see if the y-values equal 0.

f (0) = 0

f (5) = 0

since the last 2 conditions are satisfied, DNE will not be an answer choice.

3) take derivative and make it equal to 0

f' (*) = 5 - 2x

0 = 5 - 2x

x = 5/2

4) at c = 5/2, f (x) satisfies rolle's theorem.