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 1 of 5: Strengthen the target inequality
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F(a)=((b2+2)(c2+2)−3)a2−6(b+c)a+2(b2+2)(c2+2)−3(b+c)2F(a)=((b^2+2)(c^2+2)-3)a^2-6(b+c)a+2(b^2+2)(c^2+2)-3(b+c)^2
Detailed analysis

It is enough to prove the stronger inequality (a squared+2)(b squared+2)(c squared+2)≥3(a+b+c) squared, because (a+b+c) squared≥3(ab+bc+ca). After moving the right side over, the stronger inequality is F(a)≥0, where F is the displayed quadratic in a.