Let f(x) = 2x3 – 3(λ + 1) x2 + 6λx – 12 where λ > 1, has local maximum at x = α and local minimum at x = β, such that β = 2α, then the value of λ is

Engineering

Mathematics

Roots of a Cubic

Question

Let f(x) = 2x³ – 3(λ + 1) x² + 6λx – 12 where λ > 1, has local maximum at x = α and local minimum at x = β, such that β = 2α, then the value of λ is

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f '(x) = 6x² – 6 (λ + 1) x + 6λ

β = 2a

$∴$ 2 (λ + 1)2 = 9 · 1 · λ

⇒ 2λ² + 4λ + 2 = 9λ

⇒ 2λ² – 5λ + 2 = 0 ⇒ λ = 2, $\frac{1}{2}$ .