Single correct option
x and y are positive real numbers with x + y = 10. What is the maximum possible value of x²y?
- A. 125
- B. 1000/9
- C. 4000/27
- D. 500/3
Show answer and solution
Answer: (C) 4000/27
Write x²y = 4 × (x/2)(x/2)(y). The three factors x/2, x/2 and y add up to x + y = 10, a constant.
By AM–GM, their product is greatest when they are equal: x/2 = y ⇒ x = 2y, so 3y = 10 ⇒ y = 10/3 and x = 20/3.
Maximum = (20/3)² × 10/3 = 400/9 × 10/3 = 4000/27 ≈ 148.1.
125 (at x = y = 5) and 1000/9 ≈ 111 are attainable but smaller; 500/3 ≈ 166.7 exceeds the maximum and is impossible.
By AM–GM, their product is greatest when they are equal: x/2 = y ⇒ x = 2y, so 3y = 10 ⇒ y = 10/3 and x = 20/3.
Maximum = (20/3)² × 10/3 = 400/9 × 10/3 = 4000/27 ≈ 148.1.
125 (at x = y = 5) and 1000/9 ≈ 111 are attainable but smaller; 500/3 ≈ 166.7 exceeds the maximum and is impossible.
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