Which of the following statement is always true? (a)If `f(x)` is increasing, then `f^(-1)(x)` is decreasing. (b)If `f(x)` is increasing, then `1/(f(x))` is also increasing. (c)If `fa n dg` are positive functions and `f` is increasing and `g` is decreasing, then `f/g` is a decreasing function. (d)If `fa n dg` are positive functions and `f` is decreasing and `g` is increasing, the `f/g` is a decreasing function.

Options

A. is always non-positive
B. is always non-negative
C. can take positive and negative values
D. none of these

Answer

A

Explanation

`I=int_(0)^(1)f(x)[g(x)-g(1-x)]dx`
`=-int_(0)^(1)f(1-x)[g(x)-g(1-x)]dx`
or `2I=int_(0)^(1)[f(x)-f(1-x)][g(x)-g(1-x)]dx`
`=2int_(0)^(1//2)[f(x)-f(1-x)].[g(x)-g(1-x)]dxle0`