You run a business drawing caricatures. You currently have 25 drawings of rap artists and 25 drawings of "Glee" actors and actresses to sell. You can continue to draw 5 pictures per day if you are not taking the time to sell what you have already. Unfortunately, one of your classmates has seen what a great business this is, and plans to start selling pictures herself. Right now you sell each caricature for $20, but you know when your friend starts to compete for sales, you will have to discount your price to stay competitive. Assuming your sales price goes down by $1 per day, how long should you continue to draw caricatures before selling, so you make the maximum profit?

### Minimums and Maximums

In real life, it is common to need to identify what combinations of values result in a maximum or minimum quantity, collectively called **extrema**. It is important to note that not all functions have extrema.

#### Minimums

Formally: The point (*c, f*(*c*)) is the minimum value of a function if *f*(*c*) ≤ *f*(*a*) for all elements *a* (*a* ≠ *c*) of the domain of *f*.

Informally: The point (*c, f*(*c*)) is the minimum if all other function values are greater than or equal to *f*(*c*).

#### Maximums

Formally: The point (*c*, *f*(*c*)) is the maximum of *f*(*x*) if *f*(*c*) ≥ *f*(*a*) for all elements a (*a* ≠ *c*) of the domain of *f*.

Informally: The point (*c*, *f*(*c*)) is the maximum if all other function values are less than or equal to *f*(*c*).

### Examples

#### Example 1

Earlier, you were given a problem about the caricature business.

You currently have 25 drawings of rap artists and 25 drawings of "Glee" actors and actresses to sell. You can continue to draw 5 pictures per day if you are not taking the time to sell what you have already. Unfortunately, one of your classmates has seen what a great business this is, and plans to start selling pictures herself. Right now you sell each caricature for $20, but you know when your friend starts to compete for sales, you will have to discount your price to stay competitive. Assuming your sales price goes down by $1 per day, how long should you continue to draw caricatures before selling, so you make the maximum profit?

First describe the number of pictures available for sale based on the number of days, starting with the original 50 and increasing by 5 per day: \begin{align*}(50 + 5d)\end{align*}.

Next set the sales price as a function of the number of days, starting at the original price of $20, decreasing by $1 per day: \begin{align*}(20 - 1d)\end{align*}.

Now multiplying the two expressions together represents the income from the number of pictures available at the current price, based on the number of days from start: \begin{align*}(50 + 5d)(20 - d)\end{align*}

Set the combined function equal to zero and solve for the intercepts: \begin{align*}(50 + 5d)(20 - d) = 0\end{align*}. This yields the zeroes of -10 and 20. Since the expression describes a parabola, midway between the x coordinates of -10 and 20 would be the vertex, representing the greatest value resulting from the combination of sales price and number of pictures: +5

The greatest profit results from selling the pictures 5 days after the start.

If you are curious what the profit would be, or how many pictures would be sold, simply replace the variable (d) with the calculated value of 5. The value of the first expression: \begin{align*}[(50 - 5(5)]\end{align*} represents the number of pictures 5 days along. The value of the second expression \begin{align*}[20 - 1 (5)]\end{align*} represents the sales price per picture on day 5. The value of the complete expression: \begin{align*}[50 - 5(5)][20 - 1(5)]\end{align*} represents total income.

#### Example 2

Determine if each function has a minimum or a maximum point.

- y = 2
*x*- 1

The graph of y = 2*x* - 1 is a line. It does not have a maximum or a minimum.

- y =
*x*^{4}

The graph of *y* = *x*^{4} has a minimum value at (0,0). It does not have a maximum.

#### Example 3

You have 100 feet of fence with which to enclose a plot of land on the side of a barn. You want the enclosed land to be a rectangle. What size rectangle should you make with the fence in order to maximize the area of the rectangular enclosure?

The plot of land will look like the picture below:

The area of the rectangular plot is the product of its length and width. We can write the area as a function of *x: A*(*x*) = *xh*. We can eliminate *h* from the equation if we consider that we have 100 feet of fence, and we write an equation about how we are using that 100 feet of fence: *x* + 2*h* = 100. (The fourth side of the rectangle does not require fence because of the barn.) We can solve this equation for *h* and substitute into the area equation:

- \begin{align*} & x+2h=100 \\ \Rightarrow & 2h=100-x \\ \Rightarrow & h=50-\frac{x}{2} \end{align*}

\begin{align*} A(x) &= xh \\ &= x \left( 50 - \frac{x} {2} \right) \\ &= 50x-\frac{x^2}{2} \end{align*}

The graph of *A*(*x*) is shown here on the interval [0,100]. Using a maximum function on a graphing utility tells us that the point (50,1250) is the maximum point. This tells us that when the rectangle’s width is 50 ft, the area is 1250 ft^{2}.

#### Example 4

What is the minimum possible surface area of a box with a square base and a fixed volume of 12 cm^{3}?

Let the length and the width of the box be *x* cm, and the height be *h* cm. We can write the volume equation as \begin{align*}x \cdot x \cdot h = x^{2}h=12\end{align*}. We can also express the surface area in terms of *x* and *h*:

- \begin{align*}\,\! \text{Surface area} = S = 4xh+2x^{2}\end{align*}

(The base and the top are squares with area = \begin{align*}x^{2}\end{align*} and the four sides are each rectangles of area equal to \begin{align*}xh\end{align*} ).

We can express the surface area as a function of *x* if we consider the volume equation and the surface area equation as a system of equations:

- \begin{align*}\begin{cases} x^2h = 12 \\ 4xh + 2x^2 = S \\ \end{cases}\end{align*}

We want to work with the surface area equation since that is what we want to minimize. It will be easier to graph and analyze surface area if we can express *S* in terms of just one other variable. Rewrite the surface area equation as a function of *x*:

First, rewrite the volume equation:

\begin{align*}x^2h = 12 \Rightarrow h = \frac{12} {x^2}\end{align*}

Now, use substitution:

\begin{align*} S(x)&=4xh+2x^{2}\\ &=4x \left( \frac{12}{x^2}\right) +2x^{2}\\ &=\frac{48}{x}+2x^{2} \end{align*}

The values of the function *S*(*x*) represent different possibilities for the surface area of the box, given that the base is a square, and given that the volume of the box is 12 cm^{3}. To identify the minimum surface area, we need to find the lowest function values for *S*(*x*).

The graph below shows the function *S*(*x*) on the interval [0,5]. By examining the graph, we can see that the lowest point is between *x* = 2 and *x* = 3. If you use a “minimum” function on a graphing utility, you will find that the minimum point is approximately (2.3, 31.4). This tells us that when the side length of the box is approximately 2.3 cm, the surface area is approximately 31.4 cm^{2}, which is the smallest it can be.

#### Example 5

A rectangle has a perimeter of 25in. Write an expression for the area of the rectangle as a function of its width ’’x’’ and graph this expression.

The area of a rectangle is \begin{align*}l \cdot w\end{align*}

The perimeter is \begin{align*}2 \cdot l + 2 \cdot w\end{align*}

Therefore we have: \begin{align*}25 = 2(l + w)\end{align*}

\begin{align*}12.5 = l + w\end{align*}

\begin{align*}12.5 - w = l\end{align*}

\begin{align*}A = x (12.5-x)\end{align*}

Using a graphing tool, like https://www.desmos.com/calculator, we have:

### Review

- What quantity should be maximized? What quantity should be minimized? You are manufacturing chairs, and it costs you a certain amount of money to make each chair. You need to determine the selling price of the chairs.
- A rectangle has area 20 in
^{2}. Write an expression for the perimeter of the rectangle as a function of its width*x*. - What dimensions of the rectangle in problem #2 will minimize its perimeter? What is the minimum perimeter? (These values will be approximations.)
- In your own words, define the term “maximum of a function.”
- Explain how you can use a graph to identify global extrema of a function.
- A rectangle has a perimeter of 24 inches. What is the maximum area the rectangle can have?
- A cylindrical canister has a volume of 30 in
^{3}. What is the radius of the canister with minimum surface area? (Volume of a cylinder is \begin{align*} V=\pi r^{2} h\end{align*} - Consider the function
*f*(*x*) =*bx*^{2}+ 7. For what values of*b*will the function have a maximum? - Consider the function \begin{align*}S(x)=\frac{48}{x}+2x^{2}\end{align*}. How can you tell that this function does not have a global maximum or minimum?
- A rectangle has perimeter
*P*. Write a function for the area of the rectangle as a function of*P*and*x*, the width of the rectangle. What do you think will be the rectangle with maximum area? - A rectangular lot beside a river is fenced on the other 3 sides with 80ft of fencing. What is the largest possible size of the lot?

Problems 12 - 15: Determine whether each function has a maximum or minimum.

- \begin{align*}y = x^2\end{align*}
- \begin{align*}y = x^3\end{align*}
- \begin{align*}y = |x|\end{align*}
- \begin{align*}y = x + 3\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.5.