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 2 of 5: Check the leading coefficient
Detailed analysis
The coefficient of a squared in F is A=(b squared+2)(c squared+2)−3=b squared c squared+2b squared+2c squared+1, which is positive for positive b,c. Therefore it remains to show that the discriminant of F is nonpositive.