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Name: Mods: Date: 3.4 Notes on Optimization Finding Global (Absolute) Maximums and Minimums: The absolute or global extrema is the: Remember to always check the: Examples: 1. Find the absolute maximum and minimum of the function 𝑓(𝑥) = (𝑥 − 2)2 on the interval [-3,4]. First, find the critical points: Now, test the critical points and endpoints: Can we apply the idea of maximums and minimums to real life situations?? Page 1 Problems where we take real life scenarios and maximize or minimize them are called: How to solve these problems: 1. Read the problem carefully. Determine your variables. 2. Write two equations. One is your constraint equation and one is your Max/Min equation. Decide what is being maximized or minimized. 3. Solve the constraint equation for on variable. Substitute this into the Max/Min equation so it is written as a function of 1 variable. 4. Determine an appropriate domain if necessary. *5. Take the first derivative and find critical points. Test these points (and the end points) to find that you have a max or min. 6. Make sure you answer the question that was asked. Examples: 1. A farmer has 2400 feet of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? What is its maximum area? Page 2 2. Find two numbers whose sum is 10 for which the sum of their squares is a minimum. 3. We want to construct a box with a square base and we have only 10 square meters of material to use in construction of the box. Assuming that all the material is used in the construction process determine the maximum volume that the box can have. Page 3 4. A manufacturer needs to make a cylindrical can that will hold 1.5 liters of liquid. Determine the dimensions of the can that will minimize the amount of material used in its construction. Page 4 3.4 Homework 1. Find where the global max and mins are on f ( x) ( x 2)( x 3) , on the intervals: [-2, 4] and (-2, 1) 2. A ball is thrown straight up in the air from ground level. Its height after t seconds is give by the function s(t ) 16t 2 50t . When does the ball reach its maximum height? What is its maximum height? 3. A farmer has 2,000 feet of fencing to enclose a pasture area. The field will be in the shape of a rectangle and will be placed against a river where there is no fencing needed. What is the largest area of field that can be created and what are its dimensions? Page 5 4. Two numbers add up to 40. Find the numbers that maximize their product. 5. A rectangle has a perimeter of 71 feet. What length and width should it have so that its area is a maximum? What is its maximum area? Page 6