Question

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

Answer 1

Answer: 7

$f(x)=\frac{3}{2}{x}^3+\frac{\lambda x^2}{3}+x+7$
$\therefore f^{\prime }(x)=\frac{9}{2}{x}^2+\frac{2\lambda x}{3}+1$
$\Rightarrow f^{\prime }(x)=\frac{1}{6}\left(27x^2+4\lambda x+6\right)$
Clearly $D>0\Rightarrow 16{\lambda }^2-\mathrm{4.6.27}>0$
$\Rightarrow 2{\lambda }^2-81>0$
$\Rightarrow |\sqrt{2}\lambda |>9$
$\Rightarrow |\lambda |>\frac{9}{\sqrt{2}}$
$\therefore$ Least positive integral value of $\lambda$ is $7$ .