The interval in which the function $f(x)=2 x^2-\log x$, for $x>0$ decreases, is
The interval in which the function $f(x)=2 x^2-\log x$, for $x>0$ decreases, is
- (2, 4)
- $\left(0, \frac{1}{4}\right)$
- $\left(\frac{1}{2}, \infty\right)$
- $\left(0, \frac{1}{2}\right)$
Solution
We have,
$
f(x)=2 x^2-\log x, x>0
$
Differentiate w.r.t ' $x$ '
$
f^{\prime}(x)=4 x-\frac{1}{x}=\frac{4 x^2-1}{x}
$
Now, $\quad f^{\prime}(x)=0$
$
4 x^2-1=0
$
$
4 x^2=1
$
$
\Rightarrow x= \pm \frac{1}{2}
$
for decreasing
$
\begin{aligned}
f^{\prime}(x) & < 0 \\
x & \in\left(\frac{-1}{2}, \frac{1}{2}\right) \text { but } x>0
\end{aligned}
$
So,
$
x \in\left(0, \frac{1}{2}\right)
$
Asked in: AP EAMCET 2018 (23 Apr Shift 1)
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