If x , y are positive real numbers and 3 x + 4 y = 5, then the largest possible value of 16 x 2 y 3 is

Question

If x , y are positive real numbers and 3 x + 4 y = 5, then the largest possible value of 16 x 2 y 3 is,

MARKS App · Accepted Answer

2 3 x 2 + 3 4 y 3 = 5 3 x 2 + 3 x 2 + 4 y 3 + 4 y 3 + 4 y 3 = 5 Using A . M ≥ G . M for 3 x 2 , 3 x 2 , 4 y 3 , 4 y 3 , 4 y 3 3 x 2 + 3 x 2 + 4 y 3 + 4 y 3 + 4 y 3 5 ≥ 3 x 2 2 4 y 3 3 1 5 3 x + 4 y 5 ≥ 3 2 4 3 2 2 3 3 x 2 y 3 1 5 1 ≥ 2 4 3 x 2 y 3 1 5 16 x 2 y 3 ≤ 3 Maximum value of 16 x 2 y 3 = 3

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