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, ∞)
• B. (-3, 3)
• C. (3, ∞)
• D. None of these

Answer 1

The correct option is C (3, ∞)

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