Question

If x1,x2,x3 and x4 are positive real numbers such that x1x2x3x4=1,then the least value of (1+x1)(1+x2)(1+x3)(1+x4) is

Answer 1

Given info : x₁, x₂, x₃ and x₄ are positive real numbers such that x₁x₂x₃x₄ = 1

To find : the value of (1 + x₁)(1 + x₂)(1 + x₃)(1 + x₄)

solution : we know, for all real positive numbers , AM ≥ GM

For x₁, x₂, x₃ and x₄

(x₁ + x₂ + x₃ + x₄)/4 ≥ (x₁x₂x₃x₄)¼

⇒(x₁ + x₂ + x₃ + x₄) ≥ 4(1)¼ = 4

now we have to do the same thing with (1 + x₁), (1 + x₂) , (1 + x₃) and (1 + x₄).

⇒{(1 + x₁) + (1 + x₂) + (1 + x₃) + (1 + x₄)}/4 ≥{(1 + x₁)(1 + x₂)(1 + x₃)(1 + x₄)}¼

⇒{(4 + (x₁ + x₂ + x₃ + x₄)}/4 ≥ {(1 + x₁)(1 + x₂)(1 + x₃)(1 + x₄)}¼

⇒(4 + 4)/4 ≥ {(1 + x₁)(1 + x₂)(1 + x₃)(1 + x₄)}¼

⇒(2)⁴ ≥ (1 + x₁)(1 + x₂)(1 + x₃)(1 + x₄)

⇒(1 + x₁)(1 + x₂)(1 + x₃)(1 + x₄) ≤ 16

Therefore the least value of (1 + x₁)(1 + x₂)(1 + x₃)(1 + x₄) is 16