Question

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

Answer: (D)

Given $g ( x )=\ln ( h ( x ))$
$g^{\prime}(x)=\frac{h^{\prime}(x)}{h(x)} $
$g ^{\prime \prime}( x )=\frac{ h ( x ) h ^{\prime \prime}( x )-\left( h ^{\prime}( x )\right)^2}{ h ^2( x )}<0 \text { (given) } $
$\therefore g ^{\prime \prime}( x )<0 \Rightarrow g ( x ) \text { is concave down }$