Question

If $f(x)=x^3+4x^2+\lambda x+1$ is monotonically decreasing function of $x$ in the largest possible interval $(-2,-2/3)$; then find $\lambda$.

Answer 1

Answer: 4.00

$f^{\prime }(x)=3x^2+8x+\lambda <0$ in the largest interval
$(-2,-2/3)$
$\therefore D=64-12\lambda >0,f^{\prime }(-2)$
$=0,f^{\prime }(-2/3)=0$
$\therefore \lambda =4$