structural:expansion, information:extraction, information:highlighting
Examples

For a practical example, assume a situation where someone has 200 {\displaystyle 200} feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where x {\displaystyle x} is the length, y {\displaystyle y} is the width, and x y {\displaystyle xy} is the area:

The derivative with respect to x {\displaystyle x} is:

Setting this equal to 0 {\displaystyle 0}

reveals that x = 50 {\displaystyle x=50} is our only critical point.Now retrieve the endpoints by determining the interval to which x {\displaystyle x} is restricted. Since width is positive, then x > 0 {\displaystyle x>0}, and since x = 100 − y {\displaystyle x=100-y}, that implies that x < 100 {\displaystyle x<100}. Plug in critical point 50 {\displaystyle 50}, as well as endpoints 0 {\displaystyle 0} and 100 {\displaystyle 100}, into x y = x (100 − x) {\displaystyle xy=x(100-x)}, and the results are 2500, 0, {\displaystyle 2500,0,} and 0 {\displaystyle 0} respectively.

Therefore, the greatest area attainable with a rectangle of 200 {\displaystyle 200} feet of fencing is 50 × 50 = 2500 {\displaystyle 50\times 50=2500}.