# Finding the Optimum

The purpose of this project is to find the values of x and y that will yield the optimum (maximum or minimum) value of a system and the optimum value of a system using algebraic and graphical methods.

Follow these ten steps (see mp1.jpg) to determine the optimum value of z and the values of x and y that yield the optimum value.

The second attachment (mp1.jpg) has a similar project aimed at finding the dimensions of a picture frame that will give the largest possible interior area.

See attached file for full problem description.

#### Solution Preview

***See the attached file for the same text. Let me know if you have any questions.***

Question 1: For the first problem, I'm just going to follow the steps that the problem spells out (the goal of this problem is to find a local maximum or minimum of f(x, y)). For the first two steps, I'm just making up equations.

1. z = f(x, y) = 5x2 + 2y

2. y = 3x + 5

3. Equation 2 is already solved for y, so we don't have to do any extra work here.

4. I'm going to choose x = 0.

5. y = 3(0) + 5 = 0 + 5 = 5

6. Plug x = 0 and y = 5 into equation 1:

f(x, y) = 5x2 + 2y

f(0, 5) = 5(0)2 + 2(5) = 0 + 10 = 10

z = 10

7. If you do steps 4 - 6 with many values of x, you get the following values of y and z:

x y z

-5 -10 105

-4 -7 66

-3 -4 37

-2 -1 18

-1 2 9

0 5 10

1 8 21

2 11 42

3 14 73

4 17 114

5 20 165

6 23 226

7 26 297

8 29 378

9 32 469

10 35 570

It looks like there is a minimum of z = 9 (you just look for the smallest value of z). This happens when x = -1. Let's look at values of x close to ...

#### Solution Summary

The solution explains how to find the extrema (max or min) of a function z = f(x, y) in a step-by-step manner. A similar method is used in an example involving finding a maximum area for a picture frame.