Applications Of Derivatives
For f(x)=2x3−9ax2+12a2x+1, where a>0. Let p and q be the points of local maximum and local minimum respectively. If p2=q, then a equals:
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About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- applications of derivatives
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
2
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Find the derivative f′(x): f′(x)=6x2−18ax+12a2 f′(x)=6(x2−3ax+2a2)=6(x−a)(x−2a).
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Identify critical points: The critical points are x=a and x=2a. Since a>0, we have a<2a.
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Determine Max/Min: The leading coefficient is positive, so the curve goes Up-Down-Up. Thus, x=a is the local Maximum (p). x=2a is the local Minimum (q).
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Apply condition p2=q: (a)2=2a a2−2a=0⟹a(a−2)=0. Since a>0, we must have a=2.
This medium difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of applications of derivatives. It appeared in the 2025 exam.
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