Show that the function f(x) = (log x) -( 2x / 2+x) is an increasing function if x>0.

df(x)/dx = (1/x) - (2(2+x) - 2x)/(2+x)²

= (1/x) - (4+2x - 2x)/(2+x)²

= (1/x) - 4/(2+x)²

=(4+4x+x²-4x)/(x((2+x)²)

= (4+x²)/(x((2+x)²)

Since x>0. df(x)/dx >0 

(f(x+h) - f(x))/h > 0

f(x+h) - f(x) > 0

f(x+h) > f(x)

Hence f(x) is increasing funcion given x>0

Regards,

Team,

TopperLearning.