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NDA 2024 Maths · Mathematics · Application of Derivatives · medium

If x>0, y>0 and x+2y=16, then the maximum value of xy is

Single correct option

If x>0x>0, y>0y>0 and x+2y=16x+2y=16, then the maximum value of xyxy is
  1. A. 3232
  2. B. 6464
  3. C. 1616
  4. D. 128128
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Answer: (A) 3232

Substitute x=16−2yx=16-2y: xy=16y−2y2xy=16y-2y^2, which has derivative 16−4y=016-4y=0 at y=4y=4, giving x=8x=8 and xy=32xy=32; the second derivative is −4<0-4<0, so it is a maximum. By AM-GM, x⋅2y≤(162)2=64x\cdot2y\le\left(\frac{16}{2}\right)^2=64, so xy≤32xy\le32. Trap: 64 is the maximum of x⋅2yx\cdot2y and not xyxy.
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