The condition that $f(x)=a x^3+b x^2+c x+d$ has no extreme value, is
The condition that $f(x)=a x^3+b x^2+c x+d$ has no extreme value, is
$b^2>3 a c$
$b^2=4 a c$
$b^2=3 a c$
$b^2 < 3 a c$
Solution
Given curve is
$
f(x)=a x^3+b x^2+c x+d
$
On differentiating w.r.t. $x$, we get
$
f^{\prime}(x)=3 a x^2+2 b x+c
$
For extremum, $f^{\prime}(x)=0$
$
\therefore \quad 3 a x^2+2 b x+c=0
$
Since, it has no extremum value
$
\begin{array}{cl}
\therefore & b^2-4 a c < 0 \\
\Rightarrow & (2 b)^2-4 \times 3 a \times c < 0 \\
\Rightarrow & 4 b^2-12 a c < 0 \\
\Rightarrow & b^2-3 a c < 0 \\
\Rightarrow & b^2 < 3 a c
\end{array}
$