Let `h(x)=f(x)-(f(x))^2+(f(x))^3` for every real number `xdot` Then (a) `h` is increasing whenever `f` is increasing (b)`h` is increasing whenever `f` is decreasing (c)`h` is decreasing whenever `f` is decreasing (d) nothing can be said in general

Options

A. h is increasing whenever is increasing
B. h is increasing whenever f is decreasing
C. h is decreasing whenever f is decreasing
D. nothing can be said in general

Answer

A, C

Explanation

We have
`"Let" "h"(x)=f(x)-{f(x)}^2+{f(x)}^3`
`h'(x)=f'(x)-2f(x)f'(x)+3{f(x)}^2f'(x)`
`rArr h'(x)=f'(x)[1-2f(x)+3{f(x)^2}]`
`rArr h'(x)=f'(x)(3y^2-2y+1),"where" y=f(x)`
Consider the quadratic expression `3y^2-2y+1` , Clearly discriminant of this quadratic expression is less than zero . So , its sign is always same as that of `y^2`i.e.positive .
`therefore h'(x)=f'(x)xxA` positive real number
`rArr`Sign of h'(x) is same as that of f'(x)
`rArr "either "h'(x)gt0 " and " f'(x)gt0 or h'(x)lt0 and f'(x)lt0 `
`rArr` h(x) and f(x) increase and decrease together.