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NCERT Class 12 · Ch 6 Miscellaneous Exercise · Q7

NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Miscellaneous Exercise Question 7

If circle circumference plus square perimeter is k, prove the area sum is least when square side equals circle diameter.
Minimum combined area: square side equals circle diameter
Minimum combined area: square side equals circle diameter

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: Square side s=2r.
  1. For k>0, 2πr+4s=k, so s=k/4−πr/2 and 0≤r≤k/(2π).
  2. Total area A=πr²+s². Differentiating with respect to r gives A′=2πr−πs=π(2r−s).
  3. A″=2π+π²/2>0, so the unique minimum occurs at s=2r. The corresponding r=k/[2(π+4)] is feasible.

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