If x and y are real numbers with x + y = 10, what is the minimum value of x² + y² + xy?

Write x² + y² + xy = (x + y)² − xy = 100 − xy.

This is smallest when xy is as large as possible. For a fixed sum of 10, xy is largest when x = y = 5, giving xy = 25.

Minimum = 100 − 25 = 75.

There is no maximum: making x large and y negative makes xy very negative.

Tip: With the sum fixed, the product peaks when the two numbers are equal (AM ≥ GM).