Function composition and inflection points

Question

Considering two functions in $\mathbb{R}$ , $f$ and $g$, both having an inflection point on the same x-coordinate, does the function $h=f \circ g$ necessarily have an inflection point on that x-coordinate?

Answer 1

Say $f’’(c)=g’’(c)=0$. Compute the second derivative of $f\circ g$: $$((f\circ g)’)’ = ( f’(g)\cdot g’)’ = f’(g)\cdot g’’ + (g’)^2\cdot f’’(g) $$ The first term vanishes on setting $x=c$, but the latter term $$(g’(c))^2\cdot f’’(g(c))$$ is not necessarily equal to zero. So no, $f\circ g$ must not have an inflection point as well at that point.