Calculus of a Single Variable, Chapter 3, 3.2, Section 3.2, Problem 9

The Rolle's theorem is applicable to the given function, only if the function is continuous and differentiable over the interval, and f(a) = f(b). Since all polynomial functions are continuous and differentiable on R, hence, the given function is continuous and differentiable on interval. Now, you need to check if f(0) = f(3).
f(0) = -0^2 + 3*0= 0
f(3) =-3^2 + 3*3 = 0
Since all the three conditions are valid, you may apply Rolle's theorem:
f'(c)(b-a) = 0
Replacing 3 for b and 0 for a, yields:
f'(c)(3-0) = 0
You need to evaluate f'(c):
f'(c) = (-c^2 + 3c)' => f'(c) = - 2c + 3
Replacing the found values in equation f'(c)(3-0) = 0
3(-2c + 3) = 0 => -6c + 9 = 0 => -6c = -9 => c = 3/2 in (0,3)
Hence, in this case, the Rolle's theorem may be applied for c = 3/2.