Discuss the extrema of `f(x)=x/(1+xtanx),x in (0,pi/2)`

Explanation

`f'(x) = (1 - x^(2)sec^(2)x)/((1 + x tan x)^(2))`
`= (sec^(2)X (cos x + x)(cos x - x))/((1 + x tan x)^(2))`
Now the sign of the derivative depends on the factor 'cos x - x' only.
To check the sign of 'cos x - x', we draw the graphs of y = x and y = cos x.

Clearly, `f' (x_(0)) = 0`
and `f'(x) gt 0 AA x in (0, x_(0))`
`f'(x) lt 0 AA x in (x_(0), pi//2)`
Thus, `x = x_(0)` is the only point of maxima for y = f(x).