Convex function, sets and which of the following are true? (NBHM-$2014$) 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. 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 $g$.
