Algebraic Constraint Math Example 4

Follow the full solution, then compare it with the other examples linked below.

medium

x + y = 10

and

x, y > 0

, what is the maximum of

xy

1
By AM-GM: $\frac{x+y}{2} \geq \sqrt{xy}$ , so $5 \geq \sqrt{xy}$ , giving $xy \leq 25$ .
2
Maximum at $x = y = 5$ : $xy = 25$ .

Answer

xy = 25

when

x = y = 5

The constraint

x + y = 10

limits the possible products

xy

. The AM-GM inequality shows that the product is maximized when the values are equal — a powerful optimization principle.

About Algebraic Constraint

A mathematical condition expressed as an equation or inequality that restricts which values the variables are allowed to take.

What constraint does [formula] impose on [formula]?

What values does [formula] constrain [formula] to avoid?

What constraint does [formula] impose on [formula]?