MCQ
The differential coefficient of $f[\log (x)]$ when $f(x) = \log x$ is
- A$x\log x$
- B${x \over {\log x}}$
- ✓${1 \over {x\log x}}$
- D${{\log x} \over x}$
$\therefore f[\log x] = \log \log x$
$f'[\log x] = \frac{1}{{\log x}}.\frac{d}{{dx}}\log x = \frac{1}{{x\log x}}$.
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$1.$ Which of the following is true?
$(A)$ $g$ is increasing on $(1, \infty)$
$(B)$ $g$ is decreasing on $(1, \infty)$
$(C)$ $g$ is increasing on $(1,2)$ and decreasing on $(2, \infty)$
$(D)$ $g$ is decreasing on $(1,2)$ and increasing on $(2, \infty)$
$2.$ Consider the statements :
$P$ : There exists some $x \in \operatorname{IR}$ such that $f(x)+2 x=2\left(1+x^2\right)$
$Q$ : There exists some $x \in \operatorname{IR}$ such that $2 f(x)+1=2 x(1+x)$ Then
$(A)$ both $P$ and $Q$ are true
$(B)$ $P$ is true and $Q$ is false
$(C)$ $P$ is false and $Q$ is true
$(D)$ both $P$ and $Q$ are false
Give the answer question $1$ and $2.$