My Math Forum prove that $gof:I \rightarrow \mathbb R$ is also convex

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 August 26th, 2014, 12:04 PM #1 Member   Joined: Mar 2013 Posts: 71 Thanks: 4 prove that $gof:I \rightarrow \mathbb R$ is also convex Good afternoon! I'm in trouble to prove the folowing problem: "Let $f:I \rightarrow \mathbb R$ and $g:J \rightarrow \mathbb R$ be convex mappings with $f(I) \subset J$, $g$ monotonic, non decreasing. a) prove that $gof:I \rightarrow \mathbb R$ is also convex. b) give another demonstration of (a), using the fact that $f$ and $g$ are twice differentiable. c) give an exemple to show that if $g$ is not non decreasing, then the result doesn't necessarily hold My solution: a) Since $f$ is convex, then for $x,y \in I$, $0 \le t \le 1, 0 \le r \le 1, t+r=1, f(tx+ry)\le tf(x)+rf(y)$. Now I use the fact that $g$ is monotonic non-decreasing, i.e, $x  Tags$gofi, convex, mathbb, prove, rightarrow

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