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[Solved] Inequality question

 
Inequalities
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June 15, 2021 12:28 pm
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(@armor)
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a,b are real numbers and a^3+b^3=2,  show  a+b\leq 2.

 

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June 15, 2021 12:28 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

Suppose \small a+b>2,  then   \small b>2-a\Rightarrow b^3>(2-a)^3=8-12a+6a^2-a^3

\small \Rightarrow a^3+b^3>8-12a+6a^2=6a^2-12a+6+2=6(a-1)^2+2>2,

a contradiction to \small a^3+b^3=2.   Hence \small a+b\leq 2.

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