Let x, y, z be positive real numbers such that ⁻¹x + ⁻¹y + ⁻¹z = 2 . If x, y, z are the roots of the cubic equation t^3 - p t^2 + q t - r = 0 , then the minimum possible value of p^2 is

Step-by-step solution

Given ⁻¹x + ⁻¹y + ⁻¹z = 2 . Using the standard conditional identity for inverse tangents summing to 2 , we have: xy + yz + zx = 1 It is given that x, y, z are the roots of the cubic equation t^3 - p t^2 + q t - r = 0 . By Vieta's formulas, the sum of the roots taken two at a time is: q = xy + yz + zx = 1 The sum of the roots is: p = x + y + z For any real numbers x, y, z , the fundamental algebraic inequality holds: (x + y + z)^2 3(xy + yz + zx) Substituting the known values into the inequality: p^2 3(1) p^2 3 Thus