## Calculus 8th Edition

$c=1$
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, $c=1$