Question

lf $x>0$ then the interval of monotonically increasing of $f\left(x\right)=2x^2-\log x$ is

• A. $(-\infty ,\frac{-1}{2})$
• B. $(\frac{-1}{2},0)$
• C. $(0,\frac{1}{2})$
• D. $(\frac{1}{2},\infty )$

Answer 1

Answer: (D)

$(\frac{1}{2},\infty )$

Given, $f\left(x\right)=2x^2-\ln x$

$f^{\prime }\left(x\right)=4x-\frac{1}{x}$

For monotonically increasing function,

$f^{\prime }\left(x\right)>0$
Hence $4x-\frac{1}{x}>0$

$4x^2-1>0$

$x^2>\frac{1}{4}$

$x>\frac{1}{2}$ and $x<\frac{-1}{2}$.

But domain of $log\left(x\right)$ is $x>0$.

Hence the only solution for the above equation is
$x>\frac{1}{2}$

$xϵ(\frac{1}{2},\infty )$