verify that the function satisfies the three hypotheses of rolles theorem on the given interval. then find…

Explanation:

Step1: Check continuity

Polynomial functions are continuous everywhere. Since $f(x)=3x^{2}-6x + 4$ is a polynomial, it is continuous on $[-1,3]$.

Step2: Check differentiability

The derivative of $f(x)$ using the power - rule $(x^n)'=nx^{n - 1}$ is $f'(x)=6x-6$. Polynomial functions are differentiable everywhere, so $f(x)$ is differentiable on $(-1,3)$.

Step3: Check $f(-1)=f(3)$

Calculate $f(-1)$: $f(-1)=3\times(-1)^{2}-6\times(-1)+4=3 + 6+4=13$. Calculate $f(3)$: $f(3)=3\times3^{2}-6\times3 + 4=3\times9-18 + 4=27-18 + 4=13$. So $f(-1)=f(3)$.

Step4: Find $c$

By Rolle's theorem, $f'(c)=0$. Set $f'(c)=6c-6 = 0$. Solve for $c$: [ \begin{align*} 6c-6&=0\ 6c&=6\ c&=1 \end{align*} ]

Answer:

$1$