Question

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

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

Answer 1

Answer: (A)

Given, $f(x)=x^3+b{x}^2+cx+d$
$\Rightarrow f^{\prime }(x)=3x^2+2bx+c$
(As we know, if $a{x}^2+bx+c>0$ for all $x$
$\Rightarrow a>0$ and $D<0$ )
Now, $D=4b^2-12c=4\left(b^2-c\right)-8c$
(where $b^2-c<0$ and $c>0)$
$\therefore D=(-ve)ie,D<0$
$\Rightarrow f^{\prime }(x)=3x^2+2bx+c>0$
for all $x\in (-\infty ,\infty ).$
(as $D<0$ and $a>0$ )
Hence, $f(x)$ is strictly increasing function.