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Class 12 Applications Of Derivatives · Very Long Answer5 Marksconstrained_optimization

Class 12 Maths · Applications Of Derivatives · Very Long Answer #60

Two real numbers sum to 5. If the sum of their cubes is least, find the sum of their squares.

✅ Step-by-Step Solution

Final Answer: 25/2
  1. Let the numbers be x and 5−x.
  2. Their cube sum is F=x³+(5−x)³=15x²−75x+125.
  3. F′=30x−75=0 gives x=5/2.
  4. F″=30>0 throughout R, so the quadratic has its unique global minimum there; both numbers are 5/2.
  5. Their square sum is 2(5/2)²=25/2.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let the numbers be x and 5−x.
  • [1 mark] Their cube sum is F=x³+(5−x)³=15x²−75x+125.
  • [1 mark] F′=30x−75=0 gives x=5/2.
  • [1 mark] F″=30>0 throughout R, so the quadratic has its unique global minimum there; both numbers are 5/2.
  • [1 mark] Their square sum is 2(5/2)²=25/2.

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