Let f be a differentiable function with range `(0,oo)` and `g(x)=(f(x))^2 -(f(x))^3 + (f(x))^4`, then

Options

A. g(x) is always increasing
B. g(x) is always decreasing
C. g(x) is increasing if f(x) is decreasing
D. g(x) is increasing if f(x) is increasing

Answer

D

Explanation

`g'(x)=2f(x).f(x)-3.(f(x))^2.f(x)+4.(f(x))^3 f(x)`
`=[2-3(f(x))+4(f(x))^2]xxf(x).f(x)`
Let `h(x)=4. (f(x))^2 -3 f(x)+2`
D=9-4.2.4=-23
`rArr` h(x) is always (+)ve
Also f(x) is always (+)ve
`rArr` g'(x) will have same sign as of f(x)
`rArr` g(x) is increasing , if f(x) is increasing .