NCERT Class 12 · Ch 6 Exercise 6.3 · Q19
NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Exercise 6.3 Question 19
Show that a square has maximum area among rectangles inscribed in a fixed circle.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: The optimal rectangle is a square, with area 2R² for circle radius R.
- Let the half-sides be u,v>0. The diagonal condition is u²+v²=R², and area A=4uv.
- Since (u−v)²≥0, 2uv≤u²+v²=R², so A≤2R², with equality exactly when u=v.
- Equivalently A(u)=4u√(R²−u²); A′=4(R²−2u²)/√(R²−u²), positive then negative at u=R/√2.
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