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 3 of 4: Handle the two signs

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In plain words

A linear function on an interval reaches its maximum at one endpoint; the sign tells us which endpoint to choose.

s\le\frac12:\ f\le s(1-s)\le\frac14<\frac7{27};\qquad s\ge\frac12:\ f\le g(s)=\frac12s^3-\frac54s^2+s

The cubic upper-bound gap on the relevant interval.

Detailed analysis

If $s\le1/2$ , the coefficient of $p$ is nonpositive, so the maximum uses $p=0$ . If $s\ge1/2$ , it is nonnegative, so the maximum uses $p=s^2/4$ , giving the displayed cubic $g$ .