MathLabs

Problem 5

Prove that (a squared plus 2)(b squared plus 2)(c squared plus 2) is at least 9(ab+bc+ca) for all positive real numbers a,b,c.
Step 5 of 5: Finish and identify equality
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(a2+2)(b2+2)(c2+2)≥3(a+b+c)2≥9(ab+bc+ca)(a^2+2)(b^2+2)(c^2+2)\ge3(a+b+c)^2\ge9(ab+bc+ca)
Detailed analysis

The stronger inequality is proved, and the second inequality is the standard sum of three squared differences. Equality throughout occurs at a=b=c=1, which is also the equality case in the official solution.