On Cauchy inequality a. B are positive numbers, verification: B / A ^ 2 + A / b ^ 2 > 1 / A + 1 / b

On Cauchy inequality a. B are positive numbers, verification: B / A ^ 2 + A / b ^ 2 > 1 / A + 1 / b

If a and B are both positive numbers, then (B / A ^ 2 + A / b ^ 2) * (1 / B + 1 / a) (from Cauchy inequality) > = [root (B / A ^ 2 * 1 / b) + root (A / b ^ 2 * 1 / a)] ^ 2 = (1 / A + 1 / b) ^ 2 divide both sides of the inequality by 1 / A + 1 / B at the same time, we know that B / A ^ 2 + A / b ^ 2 > = 1 / A + 1 / b equal sign if and only if a = B, we get