GATE AG 2008 • Q25
MCQ • 2 Marks
Section 1: Engineering Mathematics
Calculus
Subtopic: Maxima and minima of function with several independent variables
Question Statement:
If Σ(x−aᵢ)² (i=1 to n) has a minima at A, then A is the arithmetic mean of the series
Options:
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(A) a₁−a₂+a₃−....+(−1)ⁿ⁺¹aₙ
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(B) 1/a₁+1/a₂+1/a₃+....+1/aₙ
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(C) 1/a₁−1/a₂+1/a₃−...+(−1)ⁿ⁺¹(1/aₙ)
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(D) a₁+a₂+a₃+....+aₙ
Step-by-Step Solution & Explanation
d/dx[Σ(x−aᵢ)²] = 2Σ(x−aᵢ) = 0 → nx = Σaᵢ → x = (Σaᵢ)/n, the arithmetic mean of a₁,a₂,...,aₙ (the series in option D, divided by n).
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