Prove : (a^2+b^2+c^2+d^2)(1/a^2+1/b^2+1/c^2+1/d^2)greater equal to 16
To prove the inequality (a^2+b^2+c^2+d^2)(1/a^2+1/b^2+1/c^2+1/d^2) ≥ 16, we can use the Cauchy-Schwarz inequality.
Cauchy-Schwarz Inequality states that for real numbers a₁, a₂, ..., an and b₁, b₂, ..., bn, the following inequality holds:
(a₁² + a₂² + ... + an²)(b₁² + b₂² + ... + bn²) ≥ (a₁b₁ + a₂b₂ + ... + anbn)²
We can apply this inequality to prove our inequality by considering the sequences:
a₁ = √(a²), a₂ = √(b²), a₃ = √(c²), a₄ = √(d²)
b₁ = 1/√(a²), b₂ = 1/√(b²), b₃ = 1/√(c²), b₄ = 1/√(d²)
This gives us:
(a^2 + b^2 + c^2 + d^2)(1/a^2 + 1/b^2 + 1/c^2 + 1/d^2) ≥ (√(a²) / √(a²)) + (√(b²) / √(b²)) + (√(c²) / √(c²)) + (√(d²) / √(d²))
Simplifying further, we get:
(a^2 + b^2 + c^2 + d^2)(1/a^2 + 1/b^2 + 1/c^2 + 1/d^2) ≥ (1 + 1 + 1 + 1)²
(a^2 + b^2 + c^2 + d^2)(1/a^2 + 1/b^2 + 1/c^2 + 1/d^2) ≥ 16
Hence, the inequality (a^2+b^2+c^2+d^2)(1/a^2+1/b^2+1/c^2+1/d^2) ≥ 16 is proved using the Cauchy-Schwarz inequality.