Quick Quiz True or False If f(c) is a local maximum of a continuous function f on an open interval (a, b), then f’(c) = 0. Justify your answer. If m is a local minimum and M is a local maximum of a continuous function f on (a, b), then m < M. Justify your answer.

Modeling and optimization Section 4.4 Modeling and optimization

Strategy for solving max-min problems Understand the problem: Read the problem carefully. Identify the information you need to solve the problem. Develop a mathematical model of the problem: Draw pictures and label the parts that are important to the problem. Introduce a variable to represent the quantity to be maximized or minimized. Using that variable, write a function whose extreme value gives the information sought.   Graph the function: Find the domain of the function. Determine what values of the variable make sense in the problem. Identify the critical points and endpoints: Find where the derivative is zero or fails to exist. Solve the mathematical model: If unsure of the result, support or confirm your solution with another method. Interpret the solution: Translate your mathematical result into the problem setting and decide whether the result makes sense.

Example 1 Find two numbers whose sum is 20 and whose product is as large as possible.

Example 2 An open-top box is to be made by cutting congruent squares of side length x from the corners of a 20- by 25-inch sheet of tin and bending up the sides. How large should the squares be to make the box hold as much as possible? What is the resulting maximum volume?

TOTD You are planning to make an open rectangular box from an 8- by 15-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this way, and what is its volume?

Quick Quiz Which of the following values is the absolute maximum of the function 𝑓 𝑥 =4𝑥− 𝑥 2 +6 on the interval [0, 4]? 2 4 6 10

Example 3 You have been asked to design a one-liter oil can shaped like a right circular cylinder. What dimensions will use the least material?

Example 4 What is the largest possible area for a right triangle whose hypotenuse is 5 cm long, and what are the dimensions?

TOTD A rectangle has its base on the x-axis and its upper two vertices on the parabola 𝑦=12− 𝑥 2 . What is the largest area the rectangle can have, and what are its dimensions?

Quick quiz If f is a continuous, decreasing function on [0, 10] with a critical point at (4, 2), which of the following statements must be false? f(10) is an absolute minimum of f on [0, 10] f(4) is neither a relative maximum nor a relative minimum. f’(4) does not exist f’(4) = 0 f’(4) < 0

Examples from economics r(x) = revenue from selling x items dr/dx = marginal revenue c(x) = the cost of producing x items dc/dx = marginal cost p(x) = r(x) – c(x) = the profit from selling x items dp/dx = marginal profit

Theorem 6: maximum profit Maximum profit (if any) occurs at a production level at which marginal revenue equals marginal cost.

Example 5 Suppose that 𝑟 𝑥 =9𝑥 and 𝑐 𝑥 = 𝑥 3 −6 𝑥 2 +15𝑥, where x represents thousands of units. Is there a production level that maximizes profit? It so, what is it?

Theorem 7: Minimizing average cost The production level (if any) at which average cost is smallest is a level at which the average cost equals the marginal cost.

Example 6 Suppose 𝑐 𝑥 = 𝑥 3 −6 𝑥 2 +15𝑥 where x represents thousands of units. Is there a production level that minimizes average cost? If so, what is it?

TOTD Suppose 𝑟 𝑥 = 𝑥 2 𝑥 2 +1 represents revenue and 𝑐 𝑥 = (𝑥−1) 3 3 − 1 3 represents cost, with x measured in thousands of units. Is there a production level that maximizes profits? If so, what is it?