Let g:(0, ) R be a differentiable function such that (x( x- x) e^… - Vidyadip

MCQ

Let $g:(0, \infty) \rightarrow R$ be a differentiable function such that $\int\left(\frac{x(\cos x-\sin x)}{e^{x}+1}+\frac{g(x)\left(e^{x}+1-x e^{x}\right)}{\left(e^{x}+1\right)^{2}}\right) d x=\frac{x g(x)}{e^{x}+1}+c$ for all $x >0$, where $c$ is an arbitrary constant. Then.

A
$g$ is decreasing in $\left(0, \frac{\pi}{4}\right)$
B
$g^{\prime}$ is increasing in $\left(0, \frac{\pi}{4}\right)$
C
$g+g^{\prime}$ is increasing in $\left(0, \frac{\pi}{2}\right)$
✓
$g - g ^{\prime}$ is increasing in $\left(0, \frac{\pi}{2}\right)$

✓

Answer

Correct option: D.

$g - g ^{\prime}$ is increasing in $\left(0, \frac{\pi}{2}\right)$

d
$\int\left(\frac{x(\cos x-\sin x)}{e^{x}+1}+\frac{g(x)\left(e^{x}+1-x e^{x}\right)}{\left(e^{x}+1\right)^{2}}\right) d x=\frac{x g(x)}{e^{x}+1}+c$

On differentiating both sides w.r.t. $x$, we get

$\left(\frac{x(\cos x-\sin x)}{e^{x}+1}+\frac{g(x)\left(e^{x}+1-x e^{x}\right.}{\left(e^{x}+1\right)^{2}}\right)$

$=\frac{\left(e^{x}+1\right)\left(g(x)+x g^{\prime}(x)\right)-e^{x} \cdot x \cdot g(x)}{\left(e^{x}+1\right)^{2}}$

$\left(e^{x}+1\right) x(\cos x-\sin x)+g(x)\left(e^{x}+1-x e^{x}\right)$

$=\left(e^{x}+1\right)\left(g(x)+x g^{\prime}(x)\right)-e^{x} \cdot x \cdot g(x)$

$\Rightarrow g^{\prime}(x)=\cos x-\sin x$

$\Rightarrow g(x)=\sin x+\cos x+C$

$g ( x )$ is increasing in $(0, \pi / 4)$

$g ^{\prime \prime}( x )=-\sin x -\cos x <0$

$\Rightarrow g ^{\prime}( x )$ is decreasing function

let $h ( x )= g ( x )+ g ^{\prime}( x )=2 \cos x + C$

$\Rightarrow h ^{\prime}( x )= g ^{\prime}( x )+ g ^{\prime \prime}( x )=-2 \sin x <0$

$\Rightarrow h$ is decreasing

let $\phi( x )= g ( x )- g ^{\prime}( x )=2 \sin x + C$

$\Rightarrow \phi^{\prime}( x )= g ^{\prime}( x )- g { }^{\prime \prime}( x )=2 \cos x >0$

$\Rightarrow \phi$ is increasing

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Given $\int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{dx}}{{1 + \sin x + \cos x}}} = ln\, 2$, then the value of the def. integral. $\int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{\sin \,x}}{{1 + \sin x + \cos x}}} \,dx$ is equal to

The eccentricity of a hyperbola passing through the points $(3, 0)$, $(3\sqrt 2 ,\;2)$ will be

Let the circle $S: 36 x^{2}+36 y^{2}-108 x+120 y+C=0$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $x-2 y=4$ and $2 x-y=5$ lies inside the circle $S$, then :

The value of the integral $\int_{-1}^2 \log _e\left(x+\sqrt{x^2+1}\right) d x$ is:

Suppose the sides of a triangle form a geometric progression with common ratio $r$. Then, $r$ lies in the interval

The solution of the differential equation, $x^2 \frac{{dy}}{{dx}} \cdot \cos \frac{1}{x} - y \sin \frac{1}{x} = - 1,$ where $y \rightarrow - 1$ as $x \rightarrow \infty$ is

If $z = \frac{{7 - i}}{{3 - 4i}}$ then ${z^{14}} = $

Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elements and $y$ denote the total number of one-one functions from the set $A$ to the set $A \times B$. Then ...... .

The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola, $\mathrm{x}^{2}=4 \mathrm{y},$ internally in the ratio $1: 2,$ is

Let $f$ be a function such that $f(x)+3 f\left(\frac{24}{x}\right)$ $=4 \mathrm{x}, \mathrm{x} \neq 0$. Then $\mathrm{f}(3)+\mathrm{f}(8)$ is equal to