maximum value over an open interval. However, we do know that a continuous function has an absolute maximum $($and absolute minimum$)$ over a closed interval. Therefore, let's consider the function $A(x)=100x-2x^2$ over the closed interval $0,50.$ If the maximum value occurs at an interior point, then we have found the value $x$ in the open interval $(0,50)$ that maximizes the area of the garden. Therefore, we consider the following problem: Maximize $A(x)=100x-2x^2$ over the interval $].$

Since $A$ is a continuous function on a closed, bounded interval, by the extreme value theorem, it has a maximum and a minimum. These extreme values occur either at endpoints or critical points. At the endpoints, $A(x)=0.$ Since the area is positive for all $x$ in the open interval $(0,50),$ the maximum must occur at a critical point. Differentiating the function $A(x),$ we obtain

$A^{\text{'}}(x)=100-4x$

Therefore, the only critical point is $x=25.$ We conclude that the maximum area must occur when $x=25.$ Then we have $y=100-2x=100-2(25)=50.$ To maximize the area of the garden, let $x=25ft$ and $y=50ft.$ The area of this garden is $1250f{t}^2.$

Note: You can find more examples in your textbook, section $4\mathrm{.}7$

Problem $1.$ Suppose you wished to build a gift box from a rectangular sheet of construction paper that measures $396$ cm by $396$ cm $.$ By cutting identical square pieces from each corner, the sides can be folded up and taped together, creating a lidless box. How large should you cut these squares so that the resulting box has maximum volume? What is the largest volume?

Problem $2.$

For the following, consider a pizzeria that sell pizzas for a revenue of $R(x)=15x$ and costs $C(x)=60+3x+\frac{1}{2}{x}^2,$ where $x$ represents the number of pizzas. How many pizzas sold maximizes the profit?

Problem $3.$ According to U$.$S$.$ postal regulations, the girth plus the length of a parcel sent by mail may not exceed $108$ inches, where the "girth" is the perimeter of the smallest end. What is the largest possible volume of a rectangular parcel with a square end that can be sent by mail? What are the dimensions of the package of the largest volume?

For will earn one M for every problem solved correctly. Please make sure you include all the details with a picture $($where it is possible$)$ like the example above.

These are due on Monday, December $2^{nd}$ at $9$ am$.$