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[Solved] Math problem needs to prove

 
Inequalities
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June 15, 2021 12:25 pm
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(@armor)
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Given a^2+b^2+c^2=1  show -\frac{1}{2}\leq ab+bc+ca\leq 1.

 

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June 15, 2021 12:25 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

\small a^2+b^2\geq 2ab,

\small b^2+c^2\geq 2bc,

\small c^2+a^2\geq 2ca,

add  them up to obtain

\small ab+bc+ca\geq a^2+b^2+c^2=1.

Since \small a^2+b^2+c^2+2(ab+bc+ca)\geq 0,  then 

\small ab+bc+ca\geq -\frac{1}{2}(a^2+b^2+c^2)=-\frac{1}{2},  thus   \small -\frac{1}{2}\leq ab+bc+ca\leq 1.

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