MathLabs

Problem 6

Let a,b,ca,b,c be the side lengths of a triangle. Prove that a2b(a−b)+b2c(b−c)+c2a(c−a)≥0a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge0. Determine when equality occurs.
Step 4 of 5: Apply Engel-form Cauchy-Schwarz
Previous stepNext step
xy3+yz3+zx3≥(xyz(x+y+z))2x+y+z=xyz(x+y+z)xy^3+yz^3+zx^3\ge\frac{\bigl(\sqrt{xyz}(x+y+z)\bigr)^2}{x+y+z}=xyz(x+y+z)
Detailed analysis

By Cauchy-Schwarz, ∑ui2/vi≥(∑ui)2/∑vi\sum u_i^2/v_i\ge(\sum u_i)^2/\sum v_i. Applying it to Step 3 gives the desired inequality, hence the original expression is nonnegative.