(a) Let $f_{i}: \mathbb{R}^{d} \rightarrow \mathbb{R}$ be a convex function for each $i=1, \ldots, m$. Show that

$$x \mapsto \max _{i=1, \ldots, m} f_{i}(x) \quad \text { and } \quad x \mapsto \sum_{i=1}^{m} f_{i}(x)$$

are both convex functions.

(b) Fix $c \in \mathbb{R}^{d}$. Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is convex, then $g: \mathbb{R}^{d} \rightarrow \mathbb{R}$ given by $g(x)=f\left(c^{T} x\right)$ is convex.

(c) Fix vectors $a_{1}, \ldots, a_{n} \in \mathbb{R}^{d}$. Let $Q: \mathbb{R}^{d} \rightarrow \mathbb{R}$ be given by

$$Q(\beta)=\sum_{i=1}^{n} \log \left(1+e^{a_{i}^{T} \beta}\right)+\sum_{j=1}^{d}\left|\beta_{j}\right|$$

Show that $Q$ is convex. [You may use any result from the course provided you state it.]