Find the solution of two inequalities 1) Let X and y be positive numbers and X + y = 1. Find the minimum value of 1 / x + 2 / Y and point out the value of X and y 2) When x is greater than 0, find the maximum value of 3x / x + 4

Find the solution of two inequalities 1) Let X and y be positive numbers and X + y = 1. Find the minimum value of 1 / x + 2 / Y and point out the value of X and y 2) When x is greater than 0, find the maximum value of 3x / x + 4

(1) 1 / x + 2 / y = (1 / x + 2 / y) * 1 = (1 / x + 2 / y) * (x + y) = 1 + 2x / y + 2 + Y / X ≥ 3 + 2 √ (2x / y * y / x) ≥ 3 + 2 √ 2 if and only if 2x / y = Y / x, the equal sign holds, that is, the minimum value of x = √ 2-1. Y = 2 - √ 21 / x + 2 / y = 3 + 2 √ 2 (2) 3x / (x ^ 2 + 4) = 3 / (x + 4 / x) ≤ 3 / [2 √ (x * 4 / x)] ≤ 3 / 4 if and only if x = 4 / X