Question

Let $S$ be the set of real values of parameter $\lambda$ for which the function $f(x)=2 x^3-3(2+\lambda) x^2+12 \lambda x$ has exactly one local maxima and exactly one local minima. Then the subset of $S$ is -

• A. $(5, \infty)$
• B. $(-4,4)$
• C. $(3,8)$
• D. $(-\infty,-1)$

Answer 1

Answer: (A), (C), (D)

$f(x)=2 x^3-3(2+\lambda) x^2+12 \lambda x$
$f^{\prime}(x)=6 x^2-6(2+\lambda) x+12 \lambda$
$D >0 $
$36(2+\lambda)^2-24.12 \cdot \lambda>0$
$\Rightarrow(\lambda-2)^2>0$
$\Rightarrow \lambda \neq 2$
so required set is option $( A , C , D )$