Question

The least positive integral value of ' $\lambda$ ' for which $f(x)=\frac{3 x^{3}}{2}+\frac{\lambda x^{2}}{3}+x+7$ has a point of maxima is

Answer 1

$f(x)=\frac{3}{2} x^{3}+\frac{\lambda x^{2}}{3}+x+7$
$\therefore f'(x)=\frac{9}{2} x^{2}+\frac{2 \lambda x}{3}+1$
$\Rightarrow f'(x)=\frac{1}{6}\left(27 x^{2}+4 \lambda x+6\right)$
Clearly $D > 0 \Rightarrow 16 \lambda^{2}-4.6 .27 > 0$
$\Rightarrow 2 \lambda^{2}-81 >0$
$\Rightarrow|\sqrt{2} \lambda| >9$
$\Rightarrow|\lambda| > \frac{9}{\sqrt{2}}$
$\therefore $ Least positive integral value of $\lambda$ is $7$ .