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Local Maxima

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Question

Let $a_{1}, a_{2}, a_{3}, a_{4}$ be real numbers such that $a_{1} + a_{2} + a_{3} + a_{4} = 0$ and $a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} = 1$ . Then the smallest possible value of the expression $(a_{1} - a_{2})^{2} + (a_{2} - a_{3})^{2} + (a_{3} - a_{4})^{2} + (a_{4} - a_{1})^{2}$ lies in the interval

A

(0, 1.5)

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B

(1.5, 2.5)

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C

(2.5, 3)

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D

(3, 3.5)

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Solution

The correct option is B $(1.5, 2.5)$
Given :
$a_{1} + a_{2} + a_{3} + a_{4} = 0$ and $a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} = 1$
Minimizing the given equation
$(a_{1} - a_{2})^{2} + (a_{2} - a_{3})^{2} + (a_{3} - a_{4})^{2} + (a_{4} - a_{1})^{2} = 2 [a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2}] - 2 [a_{1} a_{2} + a_{2} a_{3} + a_{3} a_{4} + a_{4} a_{1}] = 2 - 2 [a_{1} a_{2} + a_{3} (a_{2} + a_{4}) + a_{4} a_{1}] = 2 - 2 [a_{1} (a_{2} + a_{4}) + a_{3} (a_{2} + a_{4})] = 2 - 2 [(a_{2} + a_{4}) (a_{1} + a_{3})] = 2 + 2 [(a_{1} + a_{3})^{2}]$
For minimum value of the given expression, $(a_{1} + a_{3}) = 0$
$∴$ the minimum value of the expression $= 2$

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Similar questions

a_{1}, a_{2}, a_{3}, a_{4}

be real numbers such that and

a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} = 1

. Then the smallest possible value of the expression

(a_{1} - a_{2})^{2} + (a_{2} - a_{2})^{2} + (a_{3} - a_{4})^{2} + (a_{4} - a_{1})^{2}

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