DeterminantHard

Question

The least value of the product xyz for which the determinant is non-negative is :

Options

A.−8
B.−1
C.−2√2
D.−16√2

Solution

≥ 0
⇒ xyz + 2 - y - x - z ≥ 0
⇒  xyz + 2 ≥ x + y + z ≥ 3 (xyz)1/3
put (xyz)1/3 = t
⇒ t3 - 3t + 2 ≥ 0
⇒  (t - 1)(t2 + t - 2) ≥ 0
⇒  (t - 1)2(t + 2) ≥ 0
⇒ t ≥ - 2
⇒ (xyz)1/3 ≥ - 2
⇒  xyz ≥  - 8

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