Applications Of Derivatives
The minimum value of f(x)=∣x−1∣+∣x−2∣+...+∣x−2025∣ occurs at:
Select the correct option:
Solution
x=1013
For a function of the form f(x)=∑∣x−ai∣ where ai are sorted increasing order, the minimum occurs at the median of the sequence ai.
Here, the sequence is 1,2,...,2025. The total number of terms is 2025, which is an odd number. The median is the 2n+1-th term. Median position =22025+1=1013.
Thus, the minimum occurs at x=1013.
🔒 Solution Hidden from View
Submit your answer to unlock the detailed step-by-step solution.
More applications of derivatives Practice Questions
The set of all values of 'a' for which f(x)=(a+2)x3−3ax2+9ax−1 decreases for all real ...
The set of all values of 'a' for which f(x)=(a+2)x3−3ax2+9ax−1 decreases for all real ...
For f(x)=2x3−9ax2+12a2x+1, where a>0. Let p and q be the points of local maximu...
For f(x)=2x3−9ax2+12a2x+1, where a>0. Let p and q be the points of local maximu...
The acute angle between the curves y=∣x2−1∣ and y=1 at their points of intersection in th...
The acute angle between the curves y=∣x2−1∣ and y=1 at their points of intersection in th...
About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- applications of derivatives
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
x=1013
For a function of the form f(x)=∑∣x−ai∣ where ai are sorted increasing order, the minimum occurs at the median of the sequence ai.
Here, the sequence is 1,2,...,2025. The total number of terms is 2025, which is an odd number. The median is the 2n+1-th term. Median position =22025+1=1013.
Thus, the minimum occurs at x=1013.
This easy difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of applications of derivatives. It appeared in the 2025 exam.
Looking for more practice? Explore all mathematics questions or browse limit, continuity and differentiability questions on RankGuru.