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Question

Find all values of  that satisfy the conclusion of Rolle's Theorem for the function  on the interval 

Answer 1

Rolle's Theorem tells us that if f is differentiable on [0,1], and f(0) = f(1), then there exists a point c in [0,1] such that f'(c) = 0.

Indeed, 0 = f(0) = f(1) = 0. Of course, f is differentiable since it is a polynomial. So, we simply set f'(c) = 0 and solve for c: f'(c) = 2c-1 = 0. So, c = 1/2.