Chapter 3.2, Problem 56E

(a)

To determine

To calculate: An interval for the function $g\left(x\right)$ over which Rolle’s Theorem can be applied and the corresponding critical number for $g$ where, the function $f$ is a continuous on $\left[a,b\right]$, differentiable on $\left(a,b\right)$, $f\left(a\right)=f\left(b\right)$, and there exists $c$ in $\left(a,b\right)$, such that $f^{\prime }\left(c\right)=0$. So, $g\left(x\right)=f\left(x\right)+k$.

(b)

To determine

To calculate: An interval for the function $g\left(x\right)$ over which Rolle’s Theorem can be applied and the corresponding critical number for $g$ where, the function $f$ is a continuous on $\left[a,b\right]$, differentiable on $\left(a,b\right)$, $f\left(a\right)=f\left(b\right)$, and there exists $c$ in $\left(a,b\right)$, such that $f^{\prime }\left(c\right)=0$ such that $g\left(x\right)=f\left(x-k\right)$.

(c)

To determine

To calculate: An interval for the function $g\left(x\right)$ over which Rolle’s Theorem can be applied and the corresponding critical number for $g$ where, The function $f$ is a continuous on $\left[a,b\right]$, differentiable on $\left(a,b\right)$, $f\left(a\right)=f\left(b\right)$, and there exists $c$ in $\left(a,b\right)$ such that $f^{\prime }\left(c\right)=0$. Therefore, $g\left(x\right)=f\left(kx\right)$.