Question

Let a, b, c be three positive real numbers such that a + b + c = 1. Let λ =min {a³ + a²bc, b³ + ab²c, c³ + abc²}. Prove that the roots of the equation x² + x + 4λ = 0 are real.

Answer 1

Explanation:

Suppose the equation x² + x + 4λ = 0 has no real roots. Then 1 - 16λ < 0. This implies that 1 – 16(a³ + a²bc) < 0, 1 – 16(b³ + ab²c) < 0, 1 – 16(c³ + abc²) < 0, 

 Observe that 

 1-16 (a³ +a² bc) < 0 

⇒  1 – 16a² (a + bc) < 0 

⇒  1 – 16a² (1 – b – c + bc) < 0