Example 1. What are the critical points of f x 1 x x, 0 x? The maximal domain of f is 0 x and we find that

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1 6. Local Etrema of Functions We continue on our quest to etract as much information as possible about a function. The more information we gather, the better we can sketch the graph of the function. This is not the only purpose, such information is essential in applications where there is a need to interpret the behaviour of functions (which usually come from the solutions of comple problems). We now have the necessary tools to study functions it is a matter of organizing the results and relating them to the graphs of the function. We shall begin withsome of the terms used to describe the important features of a function. Several more will be introduced in this chapter. Definition A function f is said to have a stationary point at a if f a. Definition The critical points of a function f areits stationarypoints together with the points whereit is not continuous or not differentiable. The idea is that the critical points compriseall the locations where f has an interesting feature. Eample. What are the critical points of f,? The maimal domain of f is and we find that f so it has a stationary point at & since f 5 and f 5. The point is also a critical point since f is not defined at. Definition A function f is said to have a local minimum at a if its value f a is less than the value of f at all nearby points. By nearby points we mean all points close to a. Definition A function f is said to have a local maimum at a if its value f a is larger than the value of f at all nearby points. Definition A function f is said to have a local etremum at a if it is either a local maimum or a local minimum. The course tet : Calculus by Anton et al., uses the term relative rather than local. A local minumum Eample. Find the local etrema of f. By completing the square, we find f f and, since f we see that f has a local minimum at. Eample. Find the local etrema of f. Again, by completing the square, we find f f and, since f we see that f has a local maimum at. A local maimum

2 It is quitedifficult, in general, to find local etrema withthesedefinitions (the plurals of maimum / minimum / etremum are maima / minima / etrema, respectively) and weneed more streamlined tools. Theseare provided by using the derivatives of f. Using information from the first derivative We observe from the graphs on the previous page that at a local minimum, f a it is also a stationary point. As increases through the value a, f changes from a decreasing function to an increasing function, i.e., f changes from negative to positive at a local minimum, f a it is also a stationary point. As increases through the value a, f changes from an increasing function function to a decreasing, i.e., f changes from positive to negative This means that we can use the table of signs introduced in the previous chapter to investigate local etrema. Eample. Find the local etrema of f. We first find the stationary points by factorizing f : f and so the stationary points are and.,,, f Direction This table tells us that f is positive to the left of the stationary point at and negative to the right of this point, so the function has a local maimum at. It also tells us that f is negative to the left of the stationary point at and positive to the right of this point, so the function has a local minimum at. The graph of the function shown below confirms these results Eample 5. Find the local maima and minima of f e /6. We first find the stationary points by factorizing f : f and so the stationary points are and Since e /6 for all values of, we need only consider the algebraic factors and the table of signs is as set out below.,,, f Direction Hence f has a local min. value of f e 6. 5 at,

3 a local ma. value f e.at The net eample is of a slightly different type since the function has a specified domain; in previous eamples the domain was alwaysinterpreted as the maimal domain. Eample 6. Find all local etrema of the function f e sin,,. We first find the stationary points f which factorizes f e Since e for all, the stationary values are given by the equation whose only solution in, is sin (solid), cos(dashed). Since sin cos for, f on this interval, so f is increasing. On the rest of the interval:, f so the function is decreasing. The picture is completed by noting that both sin and e are positive on the domain. (N.B. sin cos f e sin,, Functions may not have local etrema It is not unusual for functions not to have local etreme. We show some eamples. Eample 7. Investigate the local etrema of the function f,. Here f f for : f is increasing f for : f is decreasing while there is no point at which f : it has no stationary points. It has a singularity at so it is the only

4 critical point. Moreover, f for all and there is no point where f : its range is,. We conclude that f has no local minimum and no local maimum: it has no local etrema. We would say that f is bounded below by f. The graph is shown below on the left Left: f, Right: f,. Eample 8. Investigate the local etrema of the function f,. Here f f for all : f is decreasing f has a singularity at (which is also a critical point). and it has no stationary points. In this case f as while f as There is no point where f : its range is R\. We conclude that f has no local etrema. We would say that f is unbounded. The graphis shown above on the right. We can now return to some of the eamples in Chapter 5 and deduce the local etrema weleave this as an eercise, the results are tabulated below. Eample Stationary point(s) local ma. local min., y 5,,9, 6,,, 7,,,5, 8 9,,,.7..., 7 7, Using information from the second derivative It the previous eamples we used the fact that the derivative of a function is positive while it is increasing and is negative while the function is decreasing. We now take this a step further. Let us suppose that f has a stationary point at a : f a. Observe that, as increases through a local minimum at a the value of the derivative changes from negative, through zero, to positive values. That is to say, the derivative f increases as passes through a local minimum. This means that its derivative must be positive i.e., f a. Similarly, as increases through a local maimum at a the value of the derivative changes from positive, through zero, to negative values. That is to say, the derivative f decreases as passes through a local maimum. This means that its derivative must be negative i.e., f a. What happens if f a? The answer is, it depends. If f at all points close to a, then we have a local minimum this is an eample of a concave (up) function, f at all points close to a, then we have a local maimum this is an eample of a

5 concave down (or conve) function f is positive on one side of a and is negative on the other, then f is said to have a point of inflection at a. These rules provide the second derivative test for localmaima & minima and points of inflection. f increasing, f, f is concave up f decreasing, f, f concave down Points of inflection Whether we use this test in preference to the table of signs approach depends on the type of function. If the derivative factorizes into a number of simple factors, the table of signs is generally preferable. Eample 9. Find all the stationary points of f and use the second derivative test to determine their nature.. Here f is differentiable throughout its domain and wehave f 6 Thus the stationary points of f are,. Since f, we find. Stationary points f 6 a local maimum - f 6 a local minimum - f at, which is an inflection point of f. Eample. Use the second derivative test to classify the local etrema of the function f e. What areits inflection points? We first find the stationary points: f e e e so f at and. To classify them, we compute the second derivative so f e e e f this is a ma / min / neither f this is a ma / min / neither The inflection points are where f. These are the roots of, which are. and.. 5

6 Stationary points (dashed) & inflection points (dotted) Eercise. Find the inflection points of the function f e sin,, (see Eample 6). Eercise. Classify the stationarypoints in Eample. Which tests arethe easiestto apply? 6