Let S be the set of real values of parameter λ for which the function f(x)=2 x3-3(2+λ) x2+12 λ x has exactly one local maxima and exactly one local minima. Then the subset of S is -

Question

Let S be the set of real values of parameter λ for which the function f(x)=2 x3-3(2+λ) x2+12 λ x has exactly one local maxima and exactly one local minima. Then the subset of S is - (A) (5, ∞) (B) (-4,4) (C) (3,8) (D) (-∞,-1)

Tardigrade Admin · Accepted Answer

f(x)=2 x3-3(2+λ) x2+12 λ x f prime(x)=6 x2-6(2+λ) x+12 λ D >0 36(2+λ)2-24.12 ⋅ λ>0 ⇒(λ-2)2>0 ⇒ λ ≠ 2 so required set is option ( A , C , D )

Q. Let $S$ be the set of real values of parameter $λ$ for which the function $f (x) = 2 x^{3} - 3 (2 + λ) x^{2} + 12 λ x$ has exactly one local maxima and exactly one local minima. Then the subset of $S$ is -

$(5, \infty)$

$(- 4, 4)$

$(3, 8)$

$(- \infty, - 1)$

$f (x) = 2 x^{3} - 3 (2 + λ) x^{2} + 12 λ x$
$f^{'} (x) = 6 x^{2} - 6 (2 + λ) x + 12 λ$
$D > 0$
$36 (2 + λ)^{2} - 24.12 \cdot λ > 0$
$\Rightarrow (λ - 2)^{2} > 0$
$\Rightarrow λ \neq = 2$
so required set is option $(A, C, D)$

Q. Let S be the set of real values of parameter λ for which the function f(x)=2x3−3(2+λ)x2+12λx has exactly one local maxima and exactly one local minima. Then the subset of S is -

(5,∞)

(−4,4)

(3,8)

(−∞,−1)