Let S be the set of real values of parameter λ for which the equatio

Question

Let S be the set of real values of parameter

λ

for which the equation

f (x) = 2 x^{3} - 3 (2 + λ) x^{2} + 12 λ x

has exactly one local maximum and exactly one local minimum. Then S is a subset of

Options:

A . (−4, ∞)

B . (−3, 3)

C . (3, ∞)

D . R

Answer: Option C
:
C

f (x) = 2 x^{3} - 3 (2 + λ) x^{2} + 12 λ x \Rightarrow f^{'} (x) = 6 x^{2} - 6 (2 + λ) x + 12 λ f^{'} (x) = 0 \Rightarrow x = 2, λ

If f(x) has exactly one local maximum and exactly one local minimum, then

λ \neq 2.

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