Question

For what values of a does the function $y=(\sqrt{3}cotx+a^2)3\times tanx$ attain its least value on the interval (0,$\pi$/2) at the point x = $\pi$/3?

Answer 1

$y=(\sqrt{3}cotx+a^2)3tanx$

$y^1=3{\sec }^{2}x(\sqrt{3}cotx+a^2)+3tanx(-\sqrt{3}{cosec}^{2}x)=0$

$\Rightarrow \sqrt{3}{cosec}^{2}xtanx={\sec }^{2}x(\sqrt{3}cotx+a^2)$

$\Rightarrow \frac{\sqrt{3}}{cosx.sinx}=\sqrt{3}\frac{cosx}{sinx}+a^2$

at $x=\frac{\pi }{3}$ ; $a^2=\sqrt{3}\left(\frac{\frac{1}{2}-\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right)=0$

$a=0$