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

Question

Let $g:(0,\infty ) \to R$ be a differentiable function such that

$\int {\left( {{{x(\cos x - \sin x)} \over {{e^x} + 1}} + {{g(x)\left( {{e^x} + 1 - x{e^x}} \right)} \over {{{({e^x} + 1)}^2}}}} \right)dx = {{x\,g(x)} \over {{e^x} + 1}} + c} $, for all x > 0, where c is an arbitrary constant. Then :

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

g' is increasing in $\left( {0,{\pi \over 4}} \right)$

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

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

Solution

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