Question

Let $h$ be a twice continuously differentiable positive function on an open interval $J$. Let $g\left(x\right)=n\left(h\right(x\left)\right)$ for each $xϵJ$.

• A. $g$ is increasing on $J$
• B. $g$ is decreasing on $J$
• C. $g$ is concave up on $J$
• D. $g$ is concave down on $J$

Answer 1

The correct option is C $g$ is concave up on $J$

$g\left(x\right)=nh\left(x\right)$

$g^{\prime \prime }\left(x\right)=n{h}^{\prime \prime }\left(x\right)$

Say $h^{\prime \prime }\left(x\right)>0$ and $h\left(x\right)>0$

Since $h\left(x\right)>0$ always then $h^{\prime \prime }\left(x\right)>0$

Hence $g^{\prime \prime }\left(x\right)>0$

$g\left(x\right)$ is concave upwards.