Let `tan^(-1)"(1-x)/(1+x)`Stament-1: The difference of the greatest and smaallest values of `f(x)" on "[0,1] is f(0)-f(1)=pi//4`
Statement-2 : `g(x)=tan^(-1)x` is an increasing functions on `[0,oo]`

Options

A. Statement-1 is True, Statement-2 is True,Statement -2 is a correct explanation for Statement -5
B. Statement -1 True ,Statement -2 is True ,Stament -2 is not a correct explanation for Statement -!
C. Statement -1 is True Statement -2 is False
D. Statement -1 is Flase,Statement -2 is True

Answer

A

Explanation

We have , `g(x)=tan^(-1)x`
`rArr g'(x)=1/(1+x^2)gt0 " for all " x rArrg(x) " is increasing on "[0,oo]`
So , statement-2 is true
Now,`f(x)=tan^(-1)1 -tan^(-1)x=pi//4-g(x)`
`therefore g(x) " is increasing on " [0,oo]`
`rArr -g(x) " is decreasing on " [0,oo]`
`rArr pi/4-g(x) " is decreasing on "[0,1]`
`rArr` f(x) is decreasing on [0,1]
`rArrf(0)=pi/4-0=pi/4` is the greatest value
and , `f(1)=pi/4-g(1)=pi/4-pi/4=0` is the least value of f(x)
Hence , required difference =`pi/4-0=pi/4`
Thus , both the statements are true and statements -2 is a correct explanation of statement-1