Let h be a twice continuously differentiable positive function on an open interval J. Let g(x)= ln (h(x)) text for each x ∈ J Suppose (h prime(x))2h prime prime(x) h(x) for each x ∈ J. Then

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 ∈ J Suppose (h prime(x))2>h prime prime(x) h(x) for each x ∈ 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

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/13fe459e49b9af9472184ff81ee3d2ed-.png" /> Given g ( x )= ln ( h ( x )) g prime(x)=(h prime(x)/h(x)) g prime prime( x )=( h ( x ) h prime prime( x )-( h prime( x ))2/ h 2( x ))<0 text (given) ∴ g prime prime( x )<0 ⇒ g ( x ) text is concave down

Q. 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

$g$ is increasing on $J$

$g$ is decreasing on $J$

$g$ is concave up on $J$

g is concave down on $J$

Given $g (x) = ln (h (x))$
$g^{'} (x) = \frac{h ^{'} ( x )}{h ( x )}$
$g^{''} (x) = \frac{h ( x ) h ^{''} ( x ) - ( h ^{'} ( x ) ) ^{2}}{h ^{2} ( x )} < 0 (given)$
$∴ g^{''} (x) < 0 \Rightarrow g (x) is concave down$

Q. Let h be a twice continuously differentiable positive function on an open interval J. Let g(x)=ln(h(x)) for each x∈J Suppose (h′(x))2>h′′(x)h(x) for each x∈J. Then

g is increasing on J

g is decreasing on J

g is concave up on J

g is concave down on J