Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 56

Suppose that all of the functions are twice differentiable and the second derivatives are never 0.

a.) If $f$ and $g$ are concave upward on interval $I$, prove that $f+g$ is concave upward on $I$.
b.) Prove that $g(x) = \left[ f(x) \right]^2$ is concave upward on $I$. Suppose that $f$ is positive and concave upward on $I$.


a.) If both $f$ and $g$ has upward concavity, $f''(x) > 0$ and $g''(x) > 0$ on $I$.
Then, $(f+g)'' = (f'+g')' = f''+g'' > 0$ on $I$.
Therefore, $(f+g)$ has upward concavity at $I$.


b.) If $f$ is positive and has upward concavity on $I$, then $f(x) > 0$ and $f''(x) > 0$

So,

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


$f(x)$ and $f''(x)$ are all positive, therefore $g(x)$ has upward concavity at interval $I$.