Using lagrange's mean value theorem, show that `(beta-alpha)/(1+beta^2) < tan^(-1)beta-tan^(-1)alpha < (beta-alpha) /(1+alpha^2) , beta > alpha > 0.`

Explanation

`"Let "f(x) =tan^(-1) x , x in [a,b] " applying LMVT"`
` f(c) =(tan^(-1) b-tan^(-1)a)/(b-a) " for a " lt c lt b " and "f(x) =(1)/(1+x^(2))`
Now f(x) is a monotonically decreasing function Hence if `a lt c lt b`
`rArr f(b) lt f(c) lt f(a)`
`rArr " "(1)/(1+b^(2)) lt(tan^(-1) b-tan^(-1)a)/(b-a) lt (1)/(1+a^(2)) " ""Hence Proved "`