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More Lessons for PreCalculus

Math Worksheets

Examples, solutions, videos, worksheets, games, and activities to help PreCalculus students learn how to find the maximum and minimum values of polynomial functions and word problems.

**Finding Maximum and Minimum Values of Polynomial Functions**

Polynomial functions are useful when solving problems that ask us to find things like maximum income, revenue or production quantities. Finding maximum and minimum values of polynomial functions help us solve these types of problems. When setting up these functions, we first determine what the problems are asking us to maximize and then set up the function accordingly.

**How we define optimization problems, and what it means to solve them?**

Solving Optimization Problems using Derivatives

Step 1: Write a function for the item to be optimized.

Step 2: Write constraint equations as needed to relate the variables. Will be used for substitution into the optimization function.

Step 3: Using substitution make the optimization function of only 1 variable.

Step 4: Find first derivative critical values and analyze to find appropriate relative max or min.

Step 5: Use the selected critical value to answer the question in the problem.

This tutorial demonstrates the solutions to 5 typical optimization problems using the first derivative to identify relative max or min values for a problem.

Example 1: Find a pair of non-negative numbers that have a product of 196 and minimize the sum of four times the first number and the second number.

Example 2: An open topped serving box will be made by cutting squares out of each corner of a 12" by 18" sheet of cardboard and folding the tabs up to form a box. What size squares should be cut out to maximize the volume of the box?

Example 3: The combined perimeter of a circle and rectangle is 100 inches. The length of the rectangle is twice its width. Find the dimensions of each in order to minimize the total area.

Example 4: What point on the function y = x^{2} - 6x + 10 is closest to the origin?

Example 5: The demand function for a product is p = $1000 - √x. The cost to produce x units is C = $50000 + $100x. What price should be set to maximize the profit of the product?

**Find the approximate maximum and minimum points of a polynomial function by graphing**

Example:

Graph f(x) = x^{3} - 4x^{2} + 5

Estimate the x-coordinates at which the relative maxima and relative minima occurs.**Optimization Problem - Maximizing the Area of Rectangular Fence Using Calculus / Derivatives**

This video shows how a farmer can find the maximum area of a rectangular pen that he can construct given 500 feet of fencing. We can actually solve this quite easily using algebra but this video tries to show the overall process that we use on maximization / minimization problems.

Example:

A man has a farm that is adjacent to a river. Suppose he wants to build a rectangular pen for his cows with 500 ft. of fencing. If one side of the pen is the river, what is the area of the largest pen he can build?**Optimization Problem - Max Volume of a Box Made From Square of Material**

Example:

How to find the maximum volume of a box made from a 2ft x 2ft piece of metal when corners of equal size are removed and then the sides of the box are folded up?

More Lessons for PreCalculus

Math Worksheets

Examples, solutions, videos, worksheets, games, and activities to help PreCalculus students learn how to find the maximum and minimum values of polynomial functions and word problems.

Polynomial functions are useful when solving problems that ask us to find things like maximum income, revenue or production quantities. Finding maximum and minimum values of polynomial functions help us solve these types of problems. When setting up these functions, we first determine what the problems are asking us to maximize and then set up the function accordingly.

Solving Optimization Problems using Derivatives

Step 1: Write a function for the item to be optimized.

Step 2: Write constraint equations as needed to relate the variables. Will be used for substitution into the optimization function.

Step 3: Using substitution make the optimization function of only 1 variable.

Step 4: Find first derivative critical values and analyze to find appropriate relative max or min.

Step 5: Use the selected critical value to answer the question in the problem.

This tutorial demonstrates the solutions to 5 typical optimization problems using the first derivative to identify relative max or min values for a problem.

Example 1: Find a pair of non-negative numbers that have a product of 196 and minimize the sum of four times the first number and the second number.

Example 2: An open topped serving box will be made by cutting squares out of each corner of a 12" by 18" sheet of cardboard and folding the tabs up to form a box. What size squares should be cut out to maximize the volume of the box?

Example 3: The combined perimeter of a circle and rectangle is 100 inches. The length of the rectangle is twice its width. Find the dimensions of each in order to minimize the total area.

Example 4: What point on the function y = x

Example 5: The demand function for a product is p = $1000 - √x. The cost to produce x units is C = $50000 + $100x. What price should be set to maximize the profit of the product?

Example:

Graph f(x) = x

Estimate the x-coordinates at which the relative maxima and relative minima occurs.

This video shows how a farmer can find the maximum area of a rectangular pen that he can construct given 500 feet of fencing. We can actually solve this quite easily using algebra but this video tries to show the overall process that we use on maximization / minimization problems.

Example:

A man has a farm that is adjacent to a river. Suppose he wants to build a rectangular pen for his cows with 500 ft. of fencing. If one side of the pen is the river, what is the area of the largest pen he can build?

Example:

How to find the maximum volume of a box made from a 2ft x 2ft piece of metal when corners of equal size are removed and then the sides of the box are folded up?

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