Question

If $f\left(x\right)=2x^3-3(2+\lambda )x^2+12\lambda x$ ($\lambda$ is real number) has exactly one local maxima and exactly one local minima and $\lambda \in R-\left\{\alpha \right\}$, then $\alpha$ is -

Answer 1

$f^{{}^{\prime }}\left(x\right)=6x^2-6(2+\lambda )x+12\lambda$

$f^{{}^{\prime }}\left(x\right)=0$ will have two real and unequal roots.

$D>0\Rightarrow (\lambda +2{)}^2-8\lambda >0$

$(\lambda -2{)}^2>0\Rightarrow \lambda \in R-\{2\}$