Problem. A farmer has 76 feet of fencing and wants to build a rectangle pen. what should the deminsions of the pen be if he wants the greatest area possible?

Solution. Denote by $a,b$ two sides of recatangle we want to maximize $ab$ undere restriction $2a+2b=76$, which is equivalent to $a+b=38$, hence $a=38-b$. Now our task is to maximize $b(38-b)$, but it is quadratic function with maximum at the point $b=19$. To sum it over, the maximum of the area attains when $a=b=19$ (when our rectangle is square).

Answer. Sides of rectangle $a=b=19$, the biggest possible area is $19^{2}=361$.