Question

If $x>0$ 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

The correct option is D $(\frac{1}{2},\infty )$

$f\left(x\right)=2x^2-\log x$

Differentiating above function w.r.t. $x$, we have

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

$f^{\prime }\left(x\right)=\frac{4x^2-1}{x}$

As we know that for an increasing function,

$f^{\prime }\left(x\right)>0$

$\frac{4x^2-1}{x}>0$

• $4x^2-1>0$

$4x^2>1$

$x>\pm \frac{1}{2}$

$x\in (-\infty ,-\frac{1}{2})\cup (\frac{1}{2},\infty ).....\left(1\right)$

• $x>0.....\left(2\right)$
  

Domain of $f\left(x\right)$ is

$x>0.....\left(3\right)$

Now, from ${eq}^n\left(1\right),\left(2\right)\&\left(3\right)$, we have

$f\left(x\right)$ is increasing for all $x\in (\frac{1}{2},\infty )$.