# Is f(x)=cot(2x)*tanx increasing or decreasing at x=pi/6?

Jun 2, 2017

$f \left(x\right)$ is decreasing at $x = \frac{\pi}{6}$

#### Explanation:

Whether a function is increasing or decreasing at a point is decided by its first derivative at that point.

As $f \left(x\right) = \cot 2 x \cdot \tan x = \frac{1}{\tan 2 x} \times \tan x$

$= \frac{1 - {\tan}^{2} x}{2 \tan x} \times 2 \tan x = \frac{1 - {\tan}^{2} x}{2} = \frac{1}{2} - \frac{1}{2} {\tan}^{2} x$

Hence, $f ' \left(x\right) = \frac{\mathrm{df}}{\mathrm{dx}} = - \frac{1}{2} \times 2 \tan x \times {\sec}^{2} x = - \tan x {\sec}^{2} x$

and $f ' \left(\frac{\pi}{6}\right) = - \tan \left(\frac{\pi}{6}\right) {\sec}^{2} \left(\frac{\pi}{6}\right) = - \frac{1}{\sqrt{3}} \times {\left(\frac{2}{\sqrt{3}}\right)}^{2} = - \frac{4}{3 \sqrt{3}}$

As $f ' \left(\frac{\pi}{6}\right) < 0$, $f \left(x\right)$ is decreasing at $x = \frac{\pi}{6}$

Observe that $x = \frac{\pi}{6} = 0.5236$ the function is decreasing in the graph below.

graph{cot(2x)*tanx [-2.5, 2.5, -1.25, 1.25]}