Problem 1

Let

x,y,z

be nonnegative real numbers with

x+y+z=1

. Show that

0\le xy+yz+zx-2xyz\le\frac{7}{27}

Step 4 of 4: Factor the final cubic

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In plain words

The double factor records the balanced point: the upper bound is attained only when all three variables are equal.

\frac7{27}-g(s)=-\frac{(3s-2)^2(6s-7)}{108}\ge0\quad(\tfrac12\le s\le1)

Detailed analysis

On $[1/2,1]$ , the square is nonnegative and $6s-7<0$ , so the factored expression is nonnegative. Equality occurs at $s=2/3$ and $p=s^2/4$ , namely $x=y=z=1/3$ .