Answer
$h(x)$ will be concave upward under the condition that $f$ is increasing.
Work Step by Step
$f''(x) \gt 0$ and $g''(x) \gt 0$ for all $x$ because they are both concave upward on $(-\infty, \infty)$.
Let $h(x) = f(g(x))$
$h'(x) = f'(g(x))~g'(x)$
$h''(x) = f''(g(x))~g'(x)~g'(x)+ f'(g(x))~g''(x)$
$h''(x) = f''(g(x))~[g'(x)]^2+ f'(g(x))~g''(x)$
Then $h''(x) \gt 0$ as long as $f'(g(x))$ is positive.
Therefore, $h(x)$ will be concave upward under the condition that $f$ is increasing.
