Mathematics - Application of Derivative Question with Solution | TestHub

$f^{\prime }(x)=4-8\sin x-\frac{\left(-\sin x+{\cos }^{2}x-{\sin }^{2}x\right)}{\cos x(1+\sin x)}+{\sec }^{2}x-2\sec x\tan x$ $=4(1-2\sin x)+{\sec }^{2}x(1-2\sin x)-\sec x(1-2\sin x)=(1-2\sin x)\left({\sec }^{2}x-\sec x+4\right)$. Since ${\sec }^{2}x-\sec x+4>0$ always, $f^{\prime }(x)>0$ iff $\sin x<\frac{1}{2}$, so $f$ is increasing in $\left(0,\frac{\pi }{6}\right)\Rightarrow a=\frac{\pi }{6}$.