Question

The function $f\left(x\right)=x^3+a{x}^2+bx+c,a^2\le 3b$ has

• A. One maximum value
• B. One minimum value
• C. No extreme value
• D. One maximum and one minimum vlue

Answer 1

The correct option is C No extreme value

Given,$f\left(x\right)=x^3++a{x}^2+bx+c,a^2\le 3b$
On differentiating w.r.t $x$, we get
$f^{\prime }\left(x\right)=3x^2+2ax+b$
Put$f^{\prime }\left(x\right)=0$
$\Rightarrow 3x^2+2ax+b=0$
$x=\frac{-2a\pm \sqrt{4a^2-12b}}{2\times 3}=\frac{-2a\pm 2\sqrt{a^2-3b}}{3}$
Since, $a^2=3b,$
$\therefore x$ has an imaginary value
Heance; no extreme value of $x$ exist.