In which interval is the function f(x) = 2x^2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $f(x) = 2x^2 - \ln |x|$.To find the interval where the function is monotonically decreasing,we need $f'(x) < 0$.First,find the derivative: $

Vedclass · Accepted Answer

$(-\infty, -1/2) \cup (0, 1/2)$

Vedclass · Answer

(C) Given $f(x) = 2x^2 - \ln |x|$.To find the interval where the function is monotonically decreasing,we need $f'(x) First,find the derivative: $f'(x) = 4x - \frac{1}{x}$.Set the derivative to be less than zero: $4x - \frac{1}{x} Simplify the expression: $\frac{4x^2 - 1}{x} Factor the numerator: $\frac{(2x - 1)(2x + 1)}{x} To solve this inequality,find the critical points by setting the numerator and denominator to zero: $x = 1/2, x = -1/2, x = 0$.Test the intervals $(-\infty, -1/2)$,$(-1/2, 0)$,$(0, 1/2)$,and $(1/2, \infty)$.For $x \in (-\infty, -1/2)$,let $x = -1$: $\frac{(2(-1)-1)(2(-1)+1)}{-1} = \frac{(-3)(-1)}{-1} = -3 For $x \in (-1/2, 0)$,let $x = -1/4$: $\frac{(2(-1/4)-1)(2(-1/4)+1)}{-1/4} = \frac{(-1.5)(0.5)}{-0.25} = 3 > 0$ (Not satisfied).For $x \in (0, 1/2)$,let $x = 1/4$: $\frac{(2(1/4)-1)(2(1/4)+1)}{1/4} = \frac{(-0.5)(1.5)}{0.25} = -3 For $x \in (1/2, \infty)$,let $x = 1$: $\frac{(2(1)-1)(2(1)+1)}{1} = \frac{(1)(3)}{1} = 3 > 0$ (Not satisfied).Thus,the function is monotonically decreasing in the interval $(-\infty, -1/2) \cup (0, 1/2)$.

In which interval is the function $f(x) = 2x^2 - \ln |x|, (x \neq 0)$ monotonically decreasing?

Similar Questions

The number of sigma $(\sigma)$ and pi $(\pi)$ bonds present in benzene respectively are:

The magnetic flux through a stationary loop with resistance $R$ varies during an interval of time $T$ as $\phi = a t(T - t)$. The heat generated during this time,neglecting the inductance of the loop,will be:

$\sum_{k=1}^{2n+1} (-1)^{k-1} k^2$ is equal to

In the electric network shown,when no current flows through the $4 \Omega$ resistor in the arm $EB$,the potential difference between the points $A$ and $D$ will be $.............\,V$.

Serum differs from blood in

Vedclass Test Series

Exam Paper Generator

Online Exam Module