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STD 12 - 6. Application of derivatives — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 6. Application of derivatives1 Mark
MCQ
Let $h$ be a twice continuously differentiable positive function on an open interval $J.$ Let $g(x) = ln\left( {h(x)} \right)$ for each $x \in J$. Suppose ${\left( {h'(x)} \right)^2} > h''(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 $J$
  • ✓
    $g$ is concave down on $J$
✓

Answer

Correct option: D.
$g$ is concave down on $J$
d
Given $g(x) = ln\left( {h(x)} \right)$

$g ' (x) = \frac{{h'(x)}}{{h(x)}}$

$g''(x) = \frac{{h(x)h''(x) - {{(h'(x))}^2}}}{{{h^2}(x)}}$ < 0

$ g''(x) \leq 0\Rightarrow  \,\,g (x)$  is concave down

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