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

Explanation

`f(x)=(1-x^(2)sec^(2)x)/(1+xtanx)^(2)=sec^(2)x(cosx+x)(cos x-x)/(1+xtanx)^(2)`
if f(x) =0 ten cos x =x
Draw the graphs of y=x and y cos x for x in `(0,(pi)/(2))`

From the figure graph intersect at `x=x_(0)`
For `x lt x_(0),cos xgt x` or `cos x- x gt 0`
For `xgtx_(0),cos x lt` or `cos x-xlt 0`
Thus derivative changes sign from positive to negative across `x=x_(0)`
so `x=x_(0)` is point of maxima.