MathLabs

Problem 1

Let x,y,zx,y,z be nonnegative real numbers with x+y+z=1x+y+z=1. Show that 0≤xy+yz+zx−2xyz≤7270\le xy+yz+zx-2xyz\le\frac{7}{27}.
Step 1 of 4: Establish the lower bound
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In plain words

The subtraction term cannot overwhelm the pairwise products: the condition that the variables add to one keeps each variable at most 11.

xy+yz+zx−2xyz=xy+z(x+y−2xy)xy+yz+zx-2xyz=xy+z(x+y-2xy)
Detailed analysis

Both terms are nonnegative: the first is clear, and the second factorizes as x(1−y)+y(1−x)x(1-y)+y(1-x) because all three variables lie in [0,1][0,1].