Skip this post if you are not a math geek.
There’s a problem in Calculus by Spivak which asks the reader to prove that if f and g are convex, and f increasing then g○f, the composition (i.e. g○f(x) = g(f(x) ) is also convex.
It’s false.
Let g(x) = 1/2 * (x^2) – 5*x
Let f(x) = exp(x*2*ln(2))
Both are convex, f increasing.
But graphing g(f(x)) shows it’s definitely not convex on R.
Addendum: It’s problem 5 in the appendix to chapter 11.
Counter example showing the graph of the composite function