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Q.
If $f ( x )= x ^{3}+4 x ^{2}+\lambda x +1$ is monotonically decreasing function of $x$ in the largest possible interval $(-2,-2 / 3)$; then find $\lambda$.
Application of Derivatives
Solution:
$f'(x)=3 x^{2}+8 x+\lambda < 0$ in the largest interval
$(-2,-2 / 3)$
$\therefore D=64-12 \lambda > 0, f'(-2)$
$=0, f'(-2 / 3)=0$
$\therefore \lambda=4$