Who can help me to solve some simple basic inequalities and Cauchy inequalities Given (x ^ 2) + 2 (y ^ 2) = 1, find the maximum value of X + 2Y Given x + y + Z = 1, find the minimum value of 2 (x ^ 2) + 3 (y ^ 2) + Z ^ 2 A B C is a real number not less than 0 to prove a ^ 3 + B ^ 3 + C ^ 3 ≥ 3ABC If a and B satisfy AB = a + B + 3, the value range of AB can be obtained

Who can help me to solve some simple basic inequalities and Cauchy inequalities Given (x ^ 2) + 2 (y ^ 2) = 1, find the maximum value of X + 2Y Given x + y + Z = 1, find the minimum value of 2 (x ^ 2) + 3 (y ^ 2) + Z ^ 2 A B C is a real number not less than 0 to prove a ^ 3 + B ^ 3 + C ^ 3 ≥ 3ABC If a and B satisfy AB = a + B + 3, the value range of AB can be obtained

The first topic: Cauchy Inequality: (1 + 2) (x ^ 2 + 2Y ^ 2) ≥ (x + 2Y) ^ 2 both sides open root sign (x ^ 2 + 2Y ^ 2) = 1, so (x + 2Y) ≤ root sign 3 the first topic: Cauchy Inequality: (1 / 2 + 1 / 3 + 1) (2x ^ 2 + 3Y ^ 2 + Z ^ 2) ≥ (x + y + Z) ^ 2 both sides divide by (1 / 2 + 1 / 3 + 1) (2x ^ 2