MCQ
In which interval $f(x) = 2x^2 - \ln |x| ,$ $(x \ne 0)$ is monotonically decreasing -
- A$(-1/2 , 1/2)$
- B$(- \infty , -1/2)$
- ✓$( - \infty , - 1/2)\, \cup \,(0,1/2)$
- D$( - \infty , - 1/2)\, \cup \,(1/2,\,\infty )$
$f^{1}(x)=4 x-\frac{1}{x}<0$
$\Longrightarrow x\left(x-\frac{1}{2}\right)\left(x+\frac{1}{2}\right)<0 \Longrightarrow x \in\left(-\infty,-\frac{1}{2}\right) \cup\left(0, \frac{1}{2}\right)$
$\ln (2), f(x)$ is monotonically decreasing
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$1.$ The correct statement$(s)$ is(are)
$(A)$ $f^{\prime}(1) < 0$
$(B)$ $f(2) < 0$
$(C)$ $f^{\prime}(x) \neq 0$ for any $x \in(1,3)$
$(D)$ $f^{\prime}(x)=0$ for some $x \in(1,3)$
$2.$ If $\int_1^3 x^2 F^{\prime}(x) d x=-12$ and $\int_1^3 x^3 F^{\prime \prime}(x) d x=40$, then the correct expression$(s)$ is(are)
$(A)$ $9 f^{\prime}(3)+f^{\prime}(1)-32=0$
$(B)$ $\int_1^3 f(x) d x=12$
$(C)$ $9 f^{\prime}(3)-f^{\prime}(1)+32=0$
$(D)$ $\int_1^3 f(x) d x=-12$
Give the answer question $1$ and $2.$