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Finding the Optimum

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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 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.

Solution Preview

**See the attached file for the same text.**

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 ...

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