MathLabs

Problem 2

Let a,b,c be positive real numbers with abc=8. Prove that a squared divided by the square root of (1+a cubed)(1+b cubed), plus the two cyclic analogues, is at least 4/3.
Step 1 of 5: Prove the one-variable lemma
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11+x3≥22+x2(x>0)\dfrac1{\sqrt{1+x^3}}\ge\dfrac2{2+x^2}\quad(x>0)
Detailed analysis

Both sides are positive. Squaring and clearing denominators gives (2+x squared) squared minus 4(1+x cubed) equal to x squared times (x-2) squared, which is nonnegative. Equality holds at x=2.