Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 58

Suppose that all of functions are twice differentiable and the second derivatives are never 0. Under what condition of $f$ will the composite function $h(x) = f(g(x))$ become upward. Suppose that $f$ and $g$ are both concave upward on $(-\infty, \infty)$.


If $f$ and $g$ are both concave upward, then $f''(x) > 0 $ and $g''(x) > 0$...
By using Chain Rule as well as Product Rule, we have...

$
\begin{equation}
\begin{aligned}
h(x) & = f(g(x))\\
\\
h'(x) &= f'(g(x))g'(x)\\
\\
h''(x) &= f''(g(x)) g''(x) \cdot g'(x) + f(g(x)) \cdot g''(x)\\
\\
h''(x) &= f''(g(x)) [g'(x)]^2 + f'(g(x)) g'' (x)
\end{aligned}
\end{equation}
$


We can say that $h''(x) > 0 $ if and only if $f'(x) > 0$.