Question
If $$f\left( x \right) = {x^3} + bx + cx + d$$ and $$0 < {b^2} < c,$$ then in $$\left( { - \infty ,\infty } \right)$$
A.
$${f\left( x \right)}$$ is a strictly increasing function
B.
$${f\left( x \right)}$$ has a local maxima
C.
$${f\left( x \right)}$$ is a strictly decreasing function
D.
$${f\left( x \right)}$$ is bounded
Answer :
$${f\left( x \right)}$$ is a strictly increasing function
Solution :
$$\eqalign{
& f\left( x \right) = {x^3} + b{x^2} + cx + d,\,0 < {b^2} < c \cr
& f'\left( x \right) = 3{x^2} + 2bx + c \cr
& {\text{Discriminant}} = 4{b^2} - 12c = 4\left( {{b^2} - 3c} \right) < 0 \cr
& \therefore \,\,f'\left( x \right) > 0\,\forall x \in R \cr
& \Rightarrow \,\,f\left( x \right)\,{\text{is}}\,{\text{strictly}}\,{\text{increasing}}\,\forall x \in R \cr} $$