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Let h(x)=f(x)-(f(x))^2+(f(x))^3 for ever...

Let `h(x)=f(x)-(f(x))^2+(f(x))^3` for every real `x`. Then,

A

h is increasing whenever f is increasing

B

h is increasing whenever f is decreasing

C

h is decreasing whenever f is increasing

D

nothing can be said in general

Text Solution

Verified by Experts

The correct Answer is:
A
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.
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