Question

The condition that $f(x)=a{x}^3+b{x}^2+cx+d$ has no extreme value is

• A. $b^2>3ac$
• B. $b^2=4ac$
• C. $b^3=3ac$
• D. $b^2<3ac$

Answer 1

Answer: (D)

$f\left(x\right)=a{x}^3+b{x}^2+cx+d$.
$f^{\prime }\left(x\right)=3a{x}^2+2bx+c$
For extreme values, $f^{\prime }\left(x\right)=0$
$\therefore 3a{x}^2+2bx+c=0$
It has no roots if $4b^2-12ac<0$
i.e., $b^2<3ac$.