Maxima, minima, critical number, extreme value theorem, closed interval method, examples and step by step solutions, local maximum and local minimum, global maximum and global minimum, fermats theorem, definition of critical number. In the problems we look at in this section, there are unknowns that we are asked to find, there is an expression involving those unknowns that must be made as. In this unit we show how differentiation can be used to find the maximum and minimum values of a function. Often this involves finding the maximum or minimum value of some. Maxima and minima problems in algebra are solved using quadratic equations of the form y fx ax. Find the most economical proportions for a covered box of fixed volume whose base is a rectangle with one side three times as long as the other. Many students find these problems intimidating because they are word problems, and because there does not appear to be a pattern to these problems. Three typical problems surface area, volume, perimeter are presented and solved here. Problems often involve multiple variables, but we can only deal with functions of one. The process of finding maximum or minimum values is called optimisation. Quadratic word problems involving maxima or minima and middle terms, respectively, in a quadratic of the form.

Examples of solving such problems without the use of calculus can be found in the. So we use our techniques for finding the maximum value of a function on a closed. We will try to tackle some questions involving maxima and minima problems. The following example has reasonably demanding algebra and involves some. Maxima and minima the diagram below shows part of a function y fx. They illustrate one of the most important applications of the first derivative. Application of maxima and minima differential calculus. The following problems are maximumminimum optimization problems. This is a part of calculus including stationary point, points of. One of the most important practical uses of higher mathematics is finding minima and maxima. If the cost per item is fixed, it is equal to the cost per item c times the number of items produced x, or cx c x. Nuffield freestanding mathematics activity maxima and minima student sheets. We want the minimum value of s, so we find the critical and singular points. These are called optimal values because they are the best possible case for the problem at hand.

In this activity you will learn how to use differentiation to find maximum and minimum. The point a is a local maximum and the point b is a local minimum. Solving problems involving cost, revenue, profit the cost function cx is the total cost of making x items. Find the maximum and minimum values of the function fx3. Use differentiation to solve the following problems. We are trying to do things like maximise the profit in a company, or minimise the costs, or find the least amount of material to make a particular object. Our approach to max min word problems is modeled after our approach to related.

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