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 4 of 5: Show the second quadratic is nonnegative
Detailed analysis
View G as a quadratic in b. Its leading coefficient is 2c squared+1>0, and its discriminant is −8(c squared−1) squared≤0. Hence G(b)≥0 for every real b, so the discriminant of F is nonpositive and F(a)≥0.