Question

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

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

Answer 1

The correct option is A $f\left(x\right)$ is strictly increasing function

$\begin{array}{c}f\left(x\right)=x^3+b{x}^2+cx+d\left(0<b^2>c\right)\\ Ifx\to \infty ,f\left(x\right)\to \infty \\ x\to -\infty ,f\left(x\right)\to -\infty \\ f^{\prime }\left(x\right)=3x^2+2bx+c\\ D\Rightarrow {\left(2b\right)}^2-4\times 3\times c\\ =4b^2-12c\\ =4\left(b^2-3c\right)\left(0<b^2>c\right)\\ =4\left(b^2-c\right)-8c\\ Now,\\ D<0\\ So,f\left(x\right)hasnotrealroots.\\ f\left(x\right)isstrictlyincrea\sin gfunction.\\ hence,optionAiscorrectanswer.\end{array}$