Answer
$c=1$
Work Step by Step
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.
$f(x) = 2x^2-4x+5$, $[-1, 3]$
$f(x)$ is a polynomial, so it is continuous on the closed interval $[-1, 3]$ and differentiable on the open interval $(-1, 3)$
$f(-1) = 11= f(3)$
Thus, all three hypotheses are satisfied. Differentiate and set the derivative $=0$ to find c.
$f'(x) = 4x-4$
$0=4(x-1)$
When $x=1$, $f'(x)=0$. Therefore, $c=1$
