Question

If $f (x) = x^3 + bx^2 + cx + d$ and $0 < b^2 < c$, then in $(- \infty , \infty)$

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

Answer 1

Answer: (A)

$f\left(x\right) = x^{3} + bx^{2}+cx+d, 0 < b^{2} < c$
$ f'\left(x\right) =3x^{2} + 2bx+c $
Discriminant $= 4b^{2} - 12c = 4 \left(b^{2} -3c\right) < 0$
$ \therefore f'\left(x\right) >0 \forall x \in R$
$ \Rightarrow $ is strictly increasing $ f\left(x\right) \forall x \in R $