A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be $\textit{convex}$ if for all $x,y \in \mathbb{R}$ and $\lambda$ such that $0 \leq \lambda \leq1,$
$f(\lambda x+ (1-\lambda)y) \leq \lambda f (x) + (1-\lambda) f(y)$.
Let $f:$$\mathbb{R}$ $→$ $\mathbb{R}$ be a convex function , and define the following functions:
$p(x) = f(-x) , \: \: \: q(x) = -f(-x), \text{ and } r(x) = f(1-x)$.
Which of the functions $p,q$ and $r$ must be convex?
- Only $p$
- Only $q$
- Only $r$
- Only $p$ and $r$
- Only $q$ and $r$