Question

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

Given, $g\left(x\right)=\ln \left(h\right(x\left)\right)$
$\Rightarrow g^{\prime }\left(x\right)=\frac{h^{\prime }\left(x\right)}{h\left(x\right)}$
$\Rightarrow g^{\prime \prime }\left(x\right)=\frac{h\left(x\right).h^{\prime \prime }\left(x\right)-\left(h^{\prime }\right(x){)}^2}{\left(h\right(x){)}^2}$
Using given condition, Clearly $g^{\prime \prime }\left(x\right)<0,xϵJ$
Hence $g\left(x\right)$ is concave down on $J.$