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[Solved] The maximum value of the expression below

 
Quadratic Equations
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June 15, 2021 10:40 am
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The real numbers  a, b, c  satisfy  a^2+b^2+c^2=9,   what is the maximum value of 

(a-b)^{2}+(b-c)^{2}+(c-a)^2 ?

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June 15, 2021 10:41 am
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

(a-b)^2+(b-c)^2+(c-a)^2=2(a^2+b^2+c^2)-(2ab+2bc+2ca)=3(a^2+b^2+c^2)-(a+b+c)^2

Since a, b, c are real numbers, (a+b+c)^2\geqslant 0.   In addition,   a^2+b^2+c^2=9.  

Thus  (a-b)^2+(b-c)^2+(c-a)^2\leqslant 3(a^2+b^2+c^2)=3*9=27.

The maximal value is 27.

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