Let x, y > 0. If x³y² = 2¹⁵, then the least value of 3x + 2y is

Solution

Maths sir · 4 steps

Break 3x + 2y into five equal-weight pieces, x, x, x, y, y, and AM-GM hands you the minimum at once.

1. Split the sum. Write 3x + 2y = x + x + x + y + y. That is five positive terms, and their product is x³y², which the question fixes at 2¹⁵.

   (x + x + x + y + y)/5 ≥ (x³y²)^(1/5) = (2¹⁵)^(1/5) = 2³ = 8

2. Lower bound. Multiply both sides by 5 to get 3x + 2y ≥ 40 for every allowed x and y.
3. Check equality. AM-GM becomes an equality when all five terms are equal, that is x = y. Then x⁵ = 2¹⁵ gives x = y = 8, and 3(8) + 2(8) = 40. So the bound is actually reached.
4. So. The least value of 3x + 2y is 40, option (d).

The idea to keep. When a product like x³y² is fixed, split the sum so each variable appears as many times as its power. Then AM-GM gives the minimum, and equal terms give the equality case.

Answer from the NTA final answer key. Solution by PrepWiser.