Let $f:]a,b[ \to\Bbb R$ be a given function. Which of the following statements are true?
a. If $f$ is convex in $]a,b[$, then the set $\tau=\{(x,y) \in\Bbb R^2| x\in ]a,b[, y\ge f(x)\}$ is a convex set.
b. If $f$ is convex in $]a,b[$, then the set $\tau=\{(x,y) \in\Bbb R^2| x\in ]a,b[, y\le f(x)\}$ is a convex set.
c. If $f$ is convex in $]a,b[$,then $|f|$ is also convex in $]a,b[$.
a) Do you know what the epigraph is? It should be if and only if, so this is true
b) No, take $f(x) = x^2$ on say $[-1, 1]$.
c) No, take $$f(x) = \left\{\begin{matrix} -x-1 & x \in[-1,0]\\ x-1 & x\in[0,1] \end{matrix}\right.$$
a. True.
c. False. Example: take $f(x) = -1 + x^2$ on $[-1, 1]$. $f''(x) = 2 > 0$ on this interval so $f$ is
convex on this interval. But $g(x) = |f(x)| = |x^2 - 1|$ is not convex on this interval. Look
at the graph of $g$.
