Largest Rectangle in Some Region in the First Quadrant

Find the area of the largest rectangle that can be inscribed in the region bounded by the curve , the line and the positive -axis.

Solution

Denote by the point as shown in the figure. Then the rectangle inscribed in the region has dimensions by . Thus, we need to maximize the area function

To this end, we compute the function’s derivative:

We see that  is the only critical point of the function.

We shall now use the first derivative test to show that the point where yields a maximum value. Note that is always positive hence the sign of depends only on the sign of the factor . It is now apparent that is positive if and negative if . By the first derivative test, the point gives a relative maximum value.

The result also follows immediately from the second derivative test. Indeed, and clearly, .

We therefore conclude that the largest rectangle that can be inscribed in the region has dimensions 1 and and its area is square units.