Question 2.10.1: An Illustration of Rolle’s Theorem Find a value of c satisfy...

First, we verify that the hypotheses of the theorem are satisfied: f is
differentiable and continuous for all x [since f(x) is a polynomial and all polynomials are continuous and differentiable everywhere]. Also, f(0) = f(1) = 2. We have

f^{\prime}(x)=3 x^2-6 x+2.

We now look for values of c such that

f^{\prime}(c)=3 c^2-6 c+2=0.

By the quadratic formula, we get t c=1+\frac{1}{3} \sqrt{3} \approx 1.5774 [not in the interval (0, 1)] and c=1-\frac{1}{3} \sqrt{3} \approx 0.42265 \in(0,1).