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NDA 2019 Maths · Mathematics · Application of Derivatives · hard

Non-negative numbers x and y satisfy x+y=9. The maximum value of x^2y is

Single correct option

Non-negative numbers xx and yy satisfy x+y=9x+y=9. The maximum value of x2yx^2y is
  1. A. 8181
  2. B. 9696
  3. C. 108108
  4. D. 162162
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Answer: (C) 108108

Substitute y=9−xy=9-x: f(x)=x2(9−x)=9x2−x3f(x)=x^2(9-x)=9x^2-x^3, f′(x)=18x−3x2=3x(6−x)f'(x)=18x-3x^2=3x(6-x). The critical point x=6x=6 gives a maximum since f′′(6)=18−36<0f''(6)=18-36<0. Then y=3y=3 and f=36⋅3=108f=36\cdot3=108. The symmetric choice x=4.5x=4.5 gives only 91.191.1, so equal splitting is not optimal for a squared term.
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