Question

Let S be the set of real values of parameter $\lambda$ for which the equation $f\left(x\right)=2x^3-3(2+\lambda )x^2+12\lambda x$ has exactly one local maximum and exactly one local minimum. Then S is a subset of

• A. $(-4,\infty )$
• B. $(-3,3)$
• C. $(3,\infty )$
• D. R

Answer 1

The correct option is C $(3,\infty )$

$f\left(x\right)=2x^3-3(2+\lambda )x^2+12\lambda x\Rightarrow f^{\prime }\left(x\right)=6x^2-6(2+\lambda )x+12\lambda f^{\prime }\left(x\right)=0\Rightarrow x=2,\lambda$
If f(x) has exactly one local maximum and exactly one local minimum, then $\lambda \ne 2.$