MA123, Chapter 6: Extreme values, Mean Value Theorem, Curve ske tching, and ConcavityChapter Goals:• Apply the Extreme Value Theorem to find the global extrema for continuous func-tion on closed and bounded interval.• Understand the connection between critical points an d local extreme values.• Understand the relationship between the sign of the derivative and the intervalson which a function is increasing and on which it is decreasing.• Understand the statement and consequences of the Mean Value Theorem.• Understand how the derivative can help you sketch the graph of a function.• Understand how to use the derivative to find the global extreme values(if any) of a continuous function over an unboun ded interval.• Understand the connection between the sign of the second derivative of a functionand the concavities of the graph of the function.• Understand the meaning of inflection points and how to locate them.Assignments:Assignment 12 Assignment 13Assignment 14 Assignment 15Finding the largest profit, or the smallest possible cost, or the shortest possible time for performing a givenprocedure or task, or figuring out how to perform a task most productively un der a given bud get and timeschedule are some examples of practical real-world applications of Calculus. The basic mathematical questionunderlying such applied problems is how to find (if they exist) the largest or sm allest values of a given functionon a given interval. This procedure depends on th e nature of the interval.◮ Global (or absolute) extre me values:The largest value a function (possibly) attains on an intervalis called its global (or absolute) maximum value . The smallest value a function (possibly) attains on aninterval is called its global (or absolute) minimum value. Both maximum and minimum values (if theyexist) are called global (or absolute) extreme values.Example 1(a):Find the maximum and minimumvalues for the functionf(x) = (x − 1)2− 3,if they exist.xyExample 1(b):Find the maximum and minimumvalues for the functionf(x) = −|x − 2| + 3,if they exist.xyExample 1(c):Find the maximum and minimumvalues for the functionf(x) = x2+ 1, x ∈ [−1, 2]if they exist.xy61We first fo cus on continuous functions on a closed and bounded interval. The question of largest and smallestvalues of a continu ou s function f on an interval that is not closed and bounded requires us to pay more attentionto the behavior of the graph of f , and specifically to where the graph is rising and where it is falling.Closed and bounded intervals:An interval is closed and bounded if it has finite length an d contains its endpoints.For example, the interval [−2, 5] is closed and bounded.◮ The Extreme Value Theorem (EVT):If a function f is continuous on a closed, bounded interval [a, b], then the fu nction f attains a maximum and aminimum value on [a, b].Example 2(a):Let f(x) =(2 +√x if x > 02 +√−x if x ≤ 0.Does f(x) have a maximum and a minimum value on [−3, 4]? Howdoes this example illustrate the Extreme Value Th eorem?xyExample 2(b): Let g(x) =1x. Does g(x) have a maximum valueand a minimum value on [−2, 3]? Does this example contradict theExtreme Value Theorem? Why or why not?xyExample 2(c): Let h(x) = x4− 2x2+ 1. Does h(x) have amaximum value and a min imum value on (−1.25, 1.5)? Does thisexample contradict the Extreme Value Theorem? Why or why not?xy62The EVT is an existence statement; it doesn’t tell you how to locate the maximum and minimum values of f.The following results tell you how to narrow down the list of possib le points on the given interval where thefunction f might have an extreme value to (usually) just a few possibilities. You can then evaluate f at thesefew possibilities, and pick out the smallest and largest value.◮ Fermat’s Theorem:Let f (x) be a continuous function on the interval [a, b]. If f has an extreme valueat a point c s trictly between a and b, and if f is differentiable at x = c, then f′(c) = 0.◮ Corollary:Let f(x) be a continuous function on the closed, bounded interval [a, b]. If f has an extremevalue at x = c in the interval, then either• c = a or c = b;• a < c < b and f′(c) = 0;• a < c < b and f is not differentiable at x = c, so that f′is not defined at x = c.Example 3:Find the maximum and minimum values of f(x) = x3−3x2−9x + 5 on the interval [0, 4].For which values x are the maximum and minimum values attained?Example 4:Find the m aximum and minimum values of F (s) =2s + 1s − 6on the interval [−1, 5]. Forwhich values s are the maximum and minimum values attained?Example 5:Find the maximum and minimum values of f (x) = x2/3on the interval [−1, 8]. For whichvalues s are the maximum and minimum values attained?63Example 6: Find th e t values on the interval [−10, 10] where g(t) = |t −4|+ 7 takes its maximum andminimum values. What are the maximum and minimum values?Example 7:Find the maximum and minimum values of k(x) =(x2+ 2x + 1 if x ≤ 1−3x + 7 if x > 1on theinterval [−2, 3].Example 8:Find the maximum and min imum values of g(x) = 1 + x + x2+ x3on the interval [0, 2].For which values x are the maximum and minimum values attained?◮ L ocal (or relative) extreme points:In addition to th e points where a function might have a maximumor minimum value, there are other points that are important for the behavior of the fu nction and the s hape ofits graph.local maxlocal minglobal maxlocal minlocal maxglobal minlocal max↑↓xyIf you think of the graph of the func-tion as the profile of a landscape, th eglobal maximum could rep resent thehighest h ill in the landscape, while theminimum could represent the deepestvalley.The other points indicated in thegraph, which look like tops of hills (al-though not the h ighest hills) and bot-tom of valleys (although not the deep-est valleys), are called local (or rela-tive) extreme values.64Definition: A function f has a local (or relative) maximum at a point (c, f(c)) if there is some intervalabout c such that f (c) ≥ f(x) for all x in that interval. A function f has a local (or relative) minimumat a point (c, f(c)) if there is some interval about c such that f(c) ≤ f (x) for all x in that interval.Theorem:If f has a local extreme value at (c, f(c)) and is differentiable at th at point c, then f′(c) = 0Critical points:Let f be a fu nction. If f is defined at the point x = c and either f′(c) = 0 or f′(c) isundefined then the point c is called a critical point of f.◮