Extreme Values o Functions When we are using mathematics to model the physical world in which we live, we oten express observed physical quantities in terms o variables. Then, unctions are used to describe the ways in which these variables change. A scientist or engineer will be interested in the ups and downs o a unction that is, its maximum and minimum values, also known as its Turning Points. Drawing a graph o a unction using a graphical calculator or computer graph plotting package will reveal this behaviour, but i we want to know the precise location o such points we need to turn to algebra and dierential calculus. In this section we look at how we can ind maximum and minimum points in this way. Consider the graph o the unction, y(x), shown below. I, at the points marked A, B and C, we draw tangents to the graph, note that these are parallel to the x axis. They are horizontal. This means that at each o the points A, B and C the slope (or gradient) o the graph is zero. The gradient o this graph is zero at each o the points A, B and C. We know that the gradient o a graph is given by dx. Consequently, dx 0 o these points are known as stationary points. at points A, B and C. All Stationary Points Any point where the tangent line to the graph is horizontal is called a stationary point. We determine stationary points by setting 0 and solving or x. dx Reerring again to above curve. Notice that at points A and B the curve actually turns. These two stationary points are reerred to as Turning Points. Point C is not a turning point because, although the graph is lat or a short time, the curve continues to go down as we look rom let to right. So, all turning points are stationary points. But not all stationary points are turning points (e.g. point C).
In other words, there are points or where 0 which are not turning points Turning Points At a Turning Point 0 dx, not all points where 0 dx dx Not all Stationary Points are Turning Points. and solving or x. are Turning Points. That is : Deinition: Absolute (Global) Extreme Values One o the most useul applications o dierential calculus is to determine i the unction assumes any maximum or minimum values on a given interval and where these values are located. Deinition: Absolute Extreme Values Let be a continuous unction with domain D. Then c is the: a) absolute maximum value on D i and only i, x c x D b) absolute minimum value on D i and only i, x c x D Absolute (or global) maximum and minimum values are also called absolute extrema. Values which are maximum or minimum to those values around them are called local extrema. Think about what happens to the slope o the graph as we travel through the minimum turning point, rom let to right, that is as x increases. As you can see the slope goes rom negative through zero to positive as x increases. In other words, you notice that or a minimum value Turning Point, as x increases, the slope dx 0 dx is increasing rom negative through zero to positive. Similarly i as x
increases the slope goes rom positive through zero to negative, the Turning Point is then a Maximum. 0 dx Deinition: The Extreme Value Theorem I is continuous on a closed interval, a minimum value on the interval a b, then has both a maximum value and Note: While you do not need to have a closed interval or a continuous unction to have an absolute maximum or minimum, the Extreme Value Theorem say that continuous unctions are guaranteed to have one on a closed bounded interval. Finding Extreme Values I a unction has a local maximum or a local minimum value at an interior point c o. its domain, and i exists at c, then c 0
Deinition: Critical Point A point in the interior o the domain o a unction at which 0 exist is called a critical point. or does not Recall: derivatives do not exist at endpoints. The deinition o Critcal Points provides us a plan to determine solutions to Extreme Value problems. 1. Determine the derivative; set derivative equal to zero; solve or variable. This gives us a set o candidates.. Locate endpoint. This gives us a set o candidates. 3. Check all candidates to see which one is the largest or smallest. Example x x Find the absolute maximum and minimum values o 3 3,. on the interval Determine Derivative 3 x x 4 x 3 3 x Critical point occurs when c 0 Critical point: 0 0 or does not exist.
Endpoints: 3 4. 16 3. 17 The absolute maximum is about 4.16 {occurs at endpoint x= 3} The absolute minimum is 0 {occurs at critical point x=0 Example Find the extreme value o x 4 x, x 1 x 1, x 1 Graphically By examining the graph we note that 1 does not exist, and 0 0 a local maximum value o 4 at x=0, and a local minimum value o at x=1.. This gives Algebraically 4 x, x x 1 1, x 1 The only point where 0 exist] is at x=0. Note: no value at x=1 [derivative does not 0 4 and 1 There are no Endpoints is this question to worry about. Thereore we see that 4 is a absolute maximum, and is a absolute minimum.
Example Find the extreme values o Graphically x x ln. x x It is hard to determine where at what are the maximums, but maybe at x 1.5 with a maximum o just about 1.1. The local minimum is at x=0, but it is not deined there. Algebraically x x x x x 0 occurs when x 0 x x x does not exist when x=0. The endpoints are, Checking these all candidate: 0 1.099 1.040 No Solution 1.040 1. 099 Thereore the absolute maximum is 1.040. The absolute minimum is 1.099.
Example Determine the maximum and minimum values o the ollowing unction with given domain x 5 1x 3 4 x, 0 x 4 and solve or x Set x 0 5 10x 1x 64x x 1 x x 10x 1x 10x 1x 10x 6 100x 3 x 4 36 3 0 3 1x 36 1x 1 3 3 The other possible values or the extreme are x=0 and x=4 Evaluate all o these in the original unction to ind the extreme values 3 16 4 17 0 3 16 4 4 5 17 Thereore the maximum value is 5 17 and the minimum value is 16.
Example Determine the where the maximum value the ollowing unction with given domain 1600v E v, 0 v 100 v 6400 occurs. Set E v 0 and solve or v v E 1600 v 6400 1600v v v 6400 1600 v 6400 v 6400 1600 80 v 80 v v 6400 The derivative=0 when numerator =0 Thereore v=-80 and v=80 But v=-80 in not part o velocity interval The other possible values or the extreme are the endpoint v=0 and v=100 Evaluate all o these in the domain to ind the extreme values E E E 0 0 80 10 100 9. 75 The maximum value is 10, which occurs at v= 80 Example The population index o a new species o pink elephants is modelled by the equation 1 P t t, 0 t 1 16t 1 population index at its lowest level?. Where t is the time in years. At what time was the Set P t 0 and solve or t Pt 16t 1 16 16t 1 16
16 0 16t 1 16 16t 1 16t 1 16t 1 16 81 16t 1 9 8 t 16 4 81 Thereore t= 4 81 The other possible values or the extreme are t=0 and t=1 Evaluate all o these in the domain to ind the extreme values P 0 1 4 P 0.098 81 1 P 1 163 Thereore the minimum value o the population index occurs at t= 4 81 years.