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If f(x)=x^3+b x^2+c x+d and 0 le b^2 le ...

If `f(x)=x^3+b x^2+c x+d` and `0 le b^2 le c ,` then a)`f(x)` is a strictly increasing function b)f(x) has local maxima c)`f(x)` is a strictly decreasing function d)`f(x)` is bounded

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

Text Solution

Verified by Experts

The correct Answer is:
A
Given, `f(x) = x ^(3) + b x ^(2) + cx + d `
`rArr " " f' (x) = 3x^(2) + 2b x + c `
As we know that, if `a x ^(2) + b x + c gt 0, AA x `, then `a gt 0` and ` Dlt 0`
Now, ` D = 4b^(2) - 12 c = 4 (b ^(2) - c) - 8c `
`" " ` [Where, `b^(2) - c lt 0 and a gt 0]`
`therefore " " D = (-"ve") or D lt 0`
`rArr f' (x) = 3x ^(2) + 2b x + c gt 0 AA x in (-oo, oo)`
`" " ` [ as D `lt` 0 and `a gt0`]
Hence, `f(x)` is strictly increasing function.
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