Let $\mathcal{C}[a, b]$ denote the vector space of continuous real-valued functions on the interval $[a, b]$, and let $\mathcal{C}^{\prime}[a, b]$ denote the subspace of continuously differentiable functions.

Show that $\|f\|_{1}=\max |f|+\max \left|f^{\prime}\right|$ defines a norm on $\mathcal{C}^{\prime}[a, b]$. Show furthermore that the map $\Phi: f \mapsto f^{\prime}((a+b) / 2)$ takes the closed unit ball $\left\{\|f\|_{1} \leqslant 1\right\} \subset \mathcal{C}^{\prime}[a, b]$ to a bounded subset of $\mathbb{R}$.

If instead we had used the norm $\|f\|_{0}=\max |f|$ restricted from $\mathcal{C}[a, b]$ to $\mathcal{C}^{\prime}[a, b]$, would $\Phi$ take the closed unit ball $\left\{\|f\|_{0} \leqslant 1\right\} \subset \mathcal{C}^{\prime}[a, b]$ to a bounded subset of $\mathbb{R}$ ? Justify your answer.