Optimization

First Derivative Test Method for finding maximum and minimum points on a function has many practical applications called Optimization - minimize cost - maximize profit - minimize material

Steps in Solving Optimization Problem 1)Understand the problem 2)Draw a diagram 3)Introduce notation 4)Combine all functions which are described 5)Use the f’ rule to find maximum or minimum

Example A manufacturing company has determined that the total cost of producing an item can be determined from the equation C = 8x x where x is the units that the company makes. How many units should the company manufacture in order to minimize the cost?

Example The product of two numbers is 150. For what two value of numbers is the sum of the first plus twice the second at a minimum?

Example A farmer has 2400 ft 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’s largest area?

Example A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions of the box would produce the maximum volume?

Example Max wants to make a box with no lid from a rectangular sheet of cardboard that is 18 inches by 24 inches. The box is to be made by cutting a square of side x from each corner of the sheet and folding the sides up. Find the value of x that maximizes the volume of the box.

Example Find the point on the parabola y 2 =2x that is closest to the point (1,4)

Example A rectangular page is to contain 24 square inches of print. The margins at the top and bottom are to be 1 ½ inches, and the margins at the right and left are to be 1 inch. What should the dimensions of the page be so that the least amount of paper is used?

Example Find the area of the largest rectangle that can be inscribed inside a semicircle of radius 4

Example A cylindrical can is to be made to hold 1000 cm 3 of oil. Find the dimensions that will minimize cost of metal in the can.