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

Options

A. If f(x) is increasing then `f^(-1)`(x) is also decreasing
B. If f(x) is increasing then `1//f(x)` is also increasing
C. If f and g are positive functions and f is increasing and g is decreasing then f/g is decreasing function
D. If f and g are positive functions and f is decreasing and g is increasing then f/g is decreasing function

Answer

4

Explanation

clearly if f(x) increases then `f^(-1)(x)` increases Draw the graph and verify
`therefore (d)/(dx)(1)/(f(x))=-(f(x))/(f^(2)(x))lt0`
`rarr (1)/(f(X))` decreases
`(d)/(dx)(f)/(g)=(f'g-f'g)/(g^(2))`
If f and g are + ve function and f lt 0 and g gt then
`(d)/(dx)(f)/(g)lt0`