Let g:(0,∞) → R Be a Function... |JEE Main 2022

JEE MAIN 2022

25-6-2022 S1

Question

Let $\mathrm{g}:(0, \infty) \rightarrow \mathrm{R}$ 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

Select the correct option:

is decreasing in $\left(0, \frac{\pi}{4}\right)$

$g^{\prime}$ is increasing in $\left(0, \frac{\pi}{4}\right)$

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

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

✓ Correct! Well done.

✗ Incorrect. Try again or view the solution.

Solution