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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?

  1. Only $p$
  2. Only $q$
  3. Only $r$
  4. Only $p$ and $r$
  5. Only $q$ and $r$
in Set Theory & Algebra by Veteran (420k points)
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