# Chapter 4 - Section 4.3 - How Derivatives Affect the Shape of a Graph - 4.3 Exercises - Page 303: 82

$h(x)$ 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.

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