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(D) Given $g(x) = \ln(h(x))$.Differentiating with respect to $x$,we get:$g'(x) = \frac{h'(x)}{h(x)}$.Differentiating again with respect to $x$ using t

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$g$ is concave down on $J$

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(D) Given $g(x) = \ln(h(x))$.Differentiating with respect to $x$,we get:$g'(x) = \frac{h'(x)}{h(x)}$.Differentiating again with respect to $x$ using the quotient rule:$g''(x) = \frac{h(x)h''(x) - (h'(x))^2}{(h(x))^2}$.We are given that $(h'(x))^2 > h''(x) h(x)$,which implies $h(x)h''(x) - (h'(x))^2 Since $(h(x))^2 > 0$ for all $x \in J$,it follows that $g''(x) = \frac{h(x)h''(x) - (h'(x))^2}{(h(x))^2} Since $g''(x) < 0$ for all $x \in J$,the function $g$ is concave down on $J$.

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

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