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I have two functions, f is monotonically increasing for every x. g has a local minimum at x=0. Now I define a new function h(x)=f(g(x)). I need to determine if h has minimum at x=0, maximum, either minimum or maximum, or if h is monotonically increasing like f is. I am not sure how to approach this. Any hints will be mostly appreciated ! Thank you.

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    As a footnote, if $f(x)=x$ for all $x,$ and $g(0)=0$ while $g(x)=x^4(\sin (1/x))^2$ for $x \ne 0,$ observe that $h(x)=g(x)$ is not monotonic on $[0,r]$ nor on $[-r,0]$ for any $r> 0,$ although $h$ is somewhat well-behaved in that it has a continuous derivative . – 2017-01-31

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$0$ will be a local minimum, here is the proof.

$g$ has a local minimum at $0$ means it exists $r$ such that for every $x \in (-r;r)$ , $g(x) \geq g(0)$.

Let $x \in (-r,r)$.

Since $f$ is monotonically increasing, for every $y,z \in \mathbb{R}$, $y \leq z \implies f(y) \leq f(z)$. Since $g(0) \leq g(x)$, then $f(g(0)) \leq f(g(x))$.

This exactly means that $h= f \circ g$ has a local minimum at 0. $h$ will not be monotonically increasing near $0$ unless you allow one function to be constant (at least near $0$ for $g$ or near $g(0)$ for $f$), because constants functions are monotonic.

Note that we have not used derivability or even continuity.

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    Great explanation, I get it now. Thank you !! – 2017-01-31