If `f(x)=x^3+bx^3+cx+d and 0 lt b^2 lt c "then in" (-oo,oo)`

Options

A. `f(x)` is strictly increasing function
B. `f(x)` has a local maxima
C. `f(x)` is strictly decreasing function
D. `f(x)` is bounded

Answer

A

Explanation

Given `f(x)=x^(3)+bx^(2)+cx+d`
`impliesf'(x)=3x^(2)+2bx+c`
(As we know if `ax^(2)+bx+cgt0` for all `x` then `agt0` and `Dlt0` in above equation)
Here `D=4b^(2)=12c=4(b^(2)-c)lt0`
[where `b^(2)-clt0` and `cgt0`]
`D=(-ve)` or `Dlt0`
`:.f'(x)=3x^(2)+2bx+cgt0` for all `xepsilon(-oo),oo)` (as `Dlt0` and `agt0`)
Hence `f(x)` is strictly increasing function.