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Class 12 Applications Of Derivatives · Very Long Answer5 Marksgeometric_optimization

Class 12 Maths · Applications Of Derivatives · Very Long Answer #109

Prove that a rectangle of greatest area inscribed in a circle of radius r>0 is a square.
Rectangle inscribed in a circle
Rectangle inscribed in a circle

✅ Step-by-Step Solution

Final Answer: Square of side r√2; Amax=2r².
  1. Let the half-sides be x,y≥0. A corner lies on the circle, giving x²+y²=r².
  2. The area A=4xy≥0 has square F=A²=16x²(r²−x²), 0≤x≤r.
  3. F′=32x(r²−2x²)=0 gives the unique interior critical point x=r/√2.
  4. F′ is positive before and negative after it, and endpoint areas vanish, so y=r/√2 and the global maximum occurs there.
  5. Both full sides equal r√2, proving it is a square; Amax=(r√2)²=2r².

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let the half-sides be x,y≥0. A corner lies on the circle, giving x²+y²=r².
  • [1 mark] The area A=4xy≥0 has square F=A²=16x²(r²−x²), 0≤x≤r.
  • [1 mark] F′=32x(r²−2x²)=0 gives the unique interior critical point x=r/√2.
  • [1 mark] F′ is positive before and negative after it, and endpoint areas vanish, so y=r/√2 and the global maximum occurs there.
  • [1 mark] Both full sides equal r√2, proving it is a square; Amax=(r√2)²=2r².

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