Question

Let S be the set of real values of parameter $\lambda$ for which the function $f\left(x\right)=2x^3-3(2+\lambda )x^2+12\lambda x$ 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)

$(3,8)$
$(-\infty ,-1)$
$(5,\infty )$

Given, $f\left(x\right)=2x^3-3(2+\lambda )x^2+12\lambda x$
$f^{\prime }\left(x\right)=6x^2-6(6+\lambda )x+12\lambda$
Now for exactly one local minima and exactly one maxima $f^{\prime }\left(x\right)$ will have exactly two distinct roots,
$\Rightarrow D>0\Rightarrow 36{(2+\lambda )}^2-24.12\lambda >0\Rightarrow {(\lambda -2)}^2>0\Rightarrow \lambda \ne 2$
so required set is option (A,C,D)