Question

Let $h$ be a twice differentiable positive function on an open interval $J.$ Let $g\left(x\right)=\ln \left(h\right(x\left)\right)$ for each $x\in J.$ Suppose ${\left(h^{\prime }\right(x\left)\right)}^2>h^{\prime \prime }\left(x\right)h\left(x\right)$ for each $x\in J.$ Then

• 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 D $g$ is concave down on $J$

$g\left(x\right)=\ln \left(h\right(x\left)\right)$
$g^{\prime }\left(x\right)=\frac{h^{\prime }\left(x\right)}{h\left(x\right)}$
Again differentiating w.r.t. $x,$
$g^{\prime \prime }\left(x\right)=\frac{h\left(x\right)h^{\prime \prime }\left(x\right)-\left(h^{\prime }\right(x){)}^2}{h^2\left(x\right)}<0$ (given)
$\therefore g^{\prime \prime }\left(x\right)<0$
$\Rightarrow g\left(x\right)$ is concave down.

$g\left(x\right)$ can be increasing or decreasing depending on the sign of $h^{\prime }\left(x\right).$