Question

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

Answer 1

`f(x)` is decreasing at `x=pi/6`

Explanation:

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

As `f(x)=cot2x*tanx=1/(tan2x) xxtanx`

`=(1-tan^2x)/(2tanx)xx2tanx=(1-tan^2x)/2=1/2-1/2tan^2x`

Hence, `f'(x)=(df)/(dx)=-1/2 xx 2tanx xx sec^2x=-tanxsec^2x`

and `f'(pi/6)=-tan(pi/6)sec^2(pi/6)=-1/sqrt3 xx (2/sqrt3)^2=-4/(3sqrt3)`

As `f'(pi/6) < 0`, `f(x)` is decreasing at `x=pi/6`

Observe that `x=pi/6=0.5236` the function is decreasing in the graph below.

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