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Class 12 Applications Of Derivatives · Very Long Answer5 Marksextrema_parameters

Class 12 Maths · Applications Of Derivatives · Very Long Answer #67

Let f(x)=2x³−9mx²+12m²x+1, m>0. It has a local maximum at x=p and a local minimum at x=q, with p²=q. Find m.

✅ Step-by-Step Solution

Final Answer: m=2
  1. f′=6x²−18mx+12m²=6(x−m)(x−2m).
  2. The stationary inputs are m and 2m, and f″=12x−18m.
  3. Since m>0, f″(m)=−6m<0 and f″(2m)=6m>0; therefore p=m and q=2m.
  4. The condition p²=q becomes m²=2m, hence m(m−2)=0.
  5. The assumption m>0 excludes 0, leaving m=2.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] f′=6x²−18mx+12m²=6(x−m)(x−2m).
  • [1 mark] The stationary inputs are m and 2m, and f″=12x−18m.
  • [1 mark] Since m>0, f″(m)=−6m<0 and f″(2m)=6m>0; therefore p=m and q=2m.
  • [1 mark] The condition p²=q becomes m²=2m, hence m(m−2)=0.
  • [1 mark] The assumption m>0 excludes 0, leaving m=2.

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