If `f(x)a n dg(x)` are two positive and increasing functions, then which of the following is not always true? (a) `[f(x)]^(g(x))` is always increasing (b) `[f(x)]^(g(x))` is decreasing, when `f(x)<1` (c) `[f(x)]^(g(x))` is increasing, then `f(x)> 1.` (d) If `f(x)>1,t h e n[f(x)]^(g(x))` is increasing.

Options

A. `[f(x)]^(g(x)` is always increasing
B. `[f(x)]^(g(x)` is decreasing when f(x) `lt` 1
C. `If[f(x)]^(g(x)` is increasing then f(x)`gt ` 1
D. If f(x) `gt` 1, then `[f(x)]^(g(x)` is increasing

Answer

1,2,3

Explanation

Let `y=f(X)^(g(x)`
`therefore (dy)/(dx)-f(x)^(g(x)[(g(x)(f(x))/(f(x))+g(x)logf(x))]`
`f(x)^(g(x),g(x),f(x)and g(X)` are positive but
log f(x) can be negative which can cause `(dy)/(dx)lt0` Hence statement (a) is false
If `f(x) lt1, then logf(x)lt0` which does not necessarily make `(dy)/(dx)lt0` hence statement (b) is false
`f(X) lt0` can also cause `(dy)/(dx)gt0` Hence statement (c ) is false but reverse of (c ) is true.