`f(x) = 5 - 12x + 3x^2, [1,3]` Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers `c` that satisfy the conclusion of Rolle’s Theorem.
Rolle's Theorem has these three hypotheses: f is continuous on [1, 3], f is differentiable on (1, 3) and f(1)=f(3).
The first and the second are obvious for a polynomial function, the third can be easily verified: f(1)=5-12+3=-4, f(3)=5-36+27=-4.
Then there exists at least one c from (1,3) such that f'(c)=0.
f'(x)=-12+6x, this is =0 for x=2 only.
The answer: hypotheses are satisfied, c=2.