Question

If the function $f(x)=2 \cot x+(2 a+1) \ln |\operatorname{cosec} x|+(2-a) x$ is strictly decreasing in $\left(0, \frac{\pi}{2}\right)$ then range of ' $a$ ' is $[\mathrm{m}, \infty)$, find the value of $\mathrm{m}$.

Answer 1

$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=-2 \operatorname{cosec}^{2} \mathrm{x}-(2 \mathrm{a}+1) \cot \mathrm{x}+(2-\mathrm{a})=-2 \cot ^{2} \mathrm{x}-(2 \mathrm{a}+1) \cot \mathrm{x}-\mathrm{a}$
$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=(\cot \mathrm{x}+\mathrm{a})(-2 \cot \mathrm{x}-1) \leq 0$ in $\left(0, \frac{\pi}{2}\right)$
$\therefore \cot \mathrm{x}+\mathrm{a} \geq 0$ in $\left(0, \frac{\pi}{2}\right)$
Hence $a \geq 0 $