Calculus Word Problems | Maxima And Minima | VolumeLocal maximum and minimum points are quite distinctive on the graph of a function, and are therefore useful in understanding the shape of the graph. In many applied problems we want to find the largest or smallest value that a function achieves for example, we might want to find the minimum cost at which some task can be performed and so identifying maximum and minimum points will be useful for applied problems as well. Some examples of local maximum and minimum points are shown in figure 5. This is important enough to state as a theorem, though we will not prove it. Theorem 5. Thus, the only points at which a function can have a local maximum or minimum are points at which the derivative is zero, as in the left hand graph in figure 5.
❖ Optimization Problem #1 ❖
Problem Set 8 Solutions.pdf
A manufacturing company has determined that the total cost of producing an item can be determined from the equation , where x is the number of units that the company makes. How many units should the company manufacture in order to minimize the cost? Solution: Take the derivative of the cost equation and set it equal to zero:. This tells us that 11 is a critical point of the equation. Now we need to figure out if this is a maximum or a minimum using the second derivative:. Since 16 is always positive, any critical value is going to be a minimum.
Problem 1: 3. Sketch the graph of a function which is continuous on [1, 5], and has all the following properties: It has an absolute maximum at 5., Exercise 1 If the monetary value of a ruby is proportional to the square of its weight, split a ruby of 2 grams in two parts so that the sum of the values of the two rubies formed is the minimal possible amount.