1. **Problem statement:** Show that the function satisfies the hypotheses of Rolle's Theorem on the interval [0,4] and find all numbers $c$ in $(0,4)$ such that $f'(c) = 0$ for $f(x) = 3x^2 - 12x + 11$. 2. **Rolle's Theorem conditions:** - $f$ is continuous on the closed interval $[a,b]$. - $f$ is differentiable on the open interval $(a,b)$. - $f(a) = f(b)$. 3. **Check continuity and differentiability:** - $f(x)$ is a polynomial, so it is continuous and differentiable everywhere, including $[0,4]$ and $(0,4)$. 4. **Check $f(0)$ and $f(4)$:** - $f(0) = 3(0)^2 - 12(0) + 11 = 11$ - $f(4) = 3(4)^2 - 12(4) + 11 = 3(16) - 48 + 11 = 48 - 48 + 11 = 11$ - Since $f(0) = f(4)$, the third condition is satisfied. 5. **Find $f'(x)$:** - $f'(x) = \frac{d}{dx}(3x^2 - 12x + 11) = 6x - 12$ 6. **Solve $f'(c) = 0$ for $c$ in $(0,4)$:** - $6c - 12 = 0$ - $6c = 12$ - $c = 2$ 7. **Conclusion:** - The function satisfies Rolle's Theorem on $[0,4]$. - The number $c$ in $(0,4)$ such that $f'(c) = 0$ is $c = 2$. **Final answer:** $c = 2$