4.3 Maximum and Minimum Values of a Function

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1 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana Maimum and Minimum Values of a Function Some of the most important applications of differential calculus are optimization problems, in which we are required to find the optimal (best) wa of doing something. Here are eamples of such problems: A farmer wants to choose the mi of crops that is likel to produce the largest profit. Adoctorwishestoselectthesmallestdosageofadrugthatwillcurecertaindisease. The manufacturer would like to minimize the cost of distributing its products. Definition 4.. A function f has an absolute maimum (or global maimum) at c if f(c) f()forallind,wheredisthedomainoff. Thenumberf(c)iscalled the maimum value of f on D. Similarl, f has an absolute minimum at c if f(c) f() for all ind andthe number f(c) is called the minimum value of f on D. The maimum and minimum values of f are called the etreme values of f. Figure 4. shows the graph of a function f with absolute maimum at b and absolute minimum at e. Note that ( b,f(b) ) is the highest point on the graph and ( e,f(e) ) is the lowest point. ( ) b,f(b) ( e,f(e) ) a b c d e Figure 4.: Maimum value f(b), minimum value f(e) In general, there is no guarantee that a function will actuall have an absolute maimum or minimum on the given interval.

2 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 84 = f() f(c) = f() c c f(c) Minimum value at c, no maimum Maimum value at c, no minimum The following theorem gives conditions under which a function is guaranteed to posses etreme values Theorem 4. (Etreme Value Theorem). If f is continuous onaclosed interval [a,b], thenf attainsanabsolutemaimum value f(c) and an minimum value f(d) at some numbers c and d in [a,b]. The Etreme Value Theorem is illustrated in the following Figure. Note that an etreme value can be taken on more than once. a c d b a c d = b a c d c b The following Figure show that a function need not be possess etreme values if either hpothesis (continuit or closed interval) is omitted from the Etreme Value Theorem. 3 + f 0 This function has minimum value f() = 0, but no maimum value. g 0 This continuous function g has no maimum and minimum. The Etreme Value Theorem sas that a continuous function on a closed interval has a maimum value and a minimum value, but it does not tell us how to find these etreme values. We start b looking for local etreme values.

3 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 85 Definition 4.. A function f has a local maimum (or relative maimum) at c if f(c) f() when is near c [This means that f(c) f() for all in some open interval containing c.] Similarl, f has a local minimum at c if f(c) f() when is near c. If f has either a local maimum or a local minimum at c, then f is said to have a local etreme values at c. Local ma [f (a) does not eist] Local ma [f (c) = 0] a b c d Local min f (b) does not eist Local min [f (d) = 0] Figure 4.: Local etreme values Figure 4. illustrates that a local etreme value can occur at a point in the domain of function at which either the graph of the function has a horizontal tangent line or function is not differentiable. Definition 4.3. A critical number of a function f is a number c in the domain of f such that f (c) = 0 or f (c) does not eist. Theorem 4.3 (Fermat s Theorem). If f has a local maimum or minimum at c, then c is a critical number of f. Fermat s Theorem does suggest that we should at least start looking for etreme values of f at a critical number of f.

4 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 86 Eample 4.6. Find the critical numbers of f() = Eample 4.7. Find the critical numbers of f() = 3/5 (4 ). Eample 4.8. Find the critical numbers of f() = +3 +.

5 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 87 To find an absolute maimum or minimum of a continuous function on a closed interval, we note that either it is local [in which case it occurs at a critical number] or it occurs at an endpoint of the interval. Thus the following three-step procedure alwas works. The Closed Interval Method To find the absolute maimum and minimum values of a continuous function f on a closed interval [a, b]:. Find the critical points of f in (a,b). Evaluate f at all the critical numbers and at the endpoints a and b. 3. The largest of the values in Step is the absolute maimum value of f on [a,b] and the smallest value is the absolute minimum. Eample 4.9. Find the absolute maimum and absolute minimum values of the function on the interval [,0]. f() = ++

6 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 88 Eample 4.0. Find the absolute maimum and absolute minimum values of the function f() = 3/ /, 9. Eercises 4.3. For each of the numbers a, b, c, d, e, r, s, and t, state whether the function whose graph is shown has an absolute maimum or minimum, a local maimum or minimum, or neither a maimum nor a minimum. (a) a b c d e r s t

7 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 89 (b) a b c d e r s t. Use the graph to state the absolute and local maimum and minimum values of the function. (a) = f() 0 (b) = f() 0 3. Find all critical numbers and the local etreme values of the following functions. (a) f() = +5 (b) f() = 3 3+ (c) f() = (d) f() = (e) f() = 3/4 4 /4 (f) f() = 3 4 (g) f() = sincos, [0,π] (h) f() = + (i) f() = (j) f() = + (e +e ) (k) f() = 4/3 +4 /3 +3 /3 (l) f() = + (m) f() = e (n) f() = sin, [0,π]

8 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana Find the absolute maimum and absolute minimum values of f on the given interval. (a) f() = 3 +5, [0,3] (b) f() = , [,] (c) f() = 4 4 +, [ 3,] (d) ) f() =, [,] (e) f() = +, [0,] (f) f() = sin+cos, [0,π/3] (g) f() = e, [0,] (h) f() = 3ln, [,4] (i) f() =, [0,3] (j) f() = 3, [,7] Answer to Eercise 4.3. (a) Absolute maimum at b; local maima at b, e, and r; absolute minimum at d; local minima at d and s (b) Absolute maimum at e; local maima at e and s; absolute minimum at t; local minima at b, c, d, r, and t. (a) Absolute maimum f(4) = 4; absolute minimum f(7) = 0 local maimum f(4) = 4, f(6) = 3; local minimum f() =, f(5) = (b) Absolute maimum f(7) = 5; absolute minimum f() = 0 local maimum f(0) =, f(3) = 4, f(5) = 3 local minimum f() = 0, f(4) =, f(6) = 3. (a) 5, absolute minimum (b), local maimum;, local minimum (c), no local etreme values (d) 0, no local etreme values; (e) 0, no local etreme values; , local minimum, local minimum (f), local minimum;, local maimum 3 (g) π, 5π, local maimum; 3π, 7π, local minimum (h), no local etreme values (i), local minimum;, local maimum (j) 0, local minimum (k),, local minimum (l), local minimum (m) 0, local maimum 3 3π (n) 0,, local minimum; π, 5π, local maimum

9 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 9 4. (a) f(0) = 5,f() = 7 (b) f() = 9,f( ) = 0 (c) f( 3) = 47,f(± ) = (d) f() = 3,f() = (e) f() =,f(0) = 0 (f) f(π/4) =,f(0) = (g) f() = /e,f(0) = 0 (h) f() =,f(3) = 3 3ln3 (i) f(3) =,f() = 0 (j) f(7) = 3,f( ) = 4.4 Increasing and Decreasing Functions Definition 4.4. Let f be defined on an interval I (open, closed, or neither), and let and denote points in I. (a) f is increasing on I if f( ) < f( ) whenever <. (b) f is decreasing on I if f( ) > f( ) whenever <. (c) f is constant on I if f( ) = f( ) for all points and. Theorem 4.4 (Increasing/ Decreasing Test). Let f be continuous on an interval I and differentiable at ever interior point of I. (i) If f () > 0 for all interior to I, then f is increasing on I. (ii) If f () < 0 for all interior to I, then f is decreasing on I. (iii) If f () = 0 for all in I, then f is constant on I. Eample 4.. Find the interval on which f() = is increasing and the interval on which it is decreasing.

10 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 9 Eample 4.. Findthe interval onwhich f() = isincreasing andthe interval onwhich + it is decreasing. Theorem 4.5 (First Derivative Test). Let f be continuous on an open interval (a,b) that contains a critical number c.. If f () > 0 for all (a,c) and f () < 0 for all (c,b), then f(c) is a local maimum value of f.. If f () < 0 for all (a,c) and f () > 0 for all (c,b), then f(c) is a local minimum value of f. 3. If f () has the same sign on both sides of c, then f(c) is not a local etreme value of f. It is eas to remember the First Derivative Test b visualizing diagram such as those in the following Figure.

11 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 93 f () > 0 f () < 0 f () < 0 f () > 0 0 c 0 c (a) Local maimum (b) Local minimum f () > 0 f () > 0 f () < 0 f () < 0 0 c 0 c (c) No maimum or minimum (d) No maimum or minimum Eample 4.3. Find the local minimum and maimum values of the function f() =

12 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 94 Eample 4.4. Findthelocalminimumandmaimumvaluesoff() ( = (sin) /3 ontheinterval π, ) π 6 3. Eercises 4.4. Find the intervals on which f is increasing or decreasing. (a) f() = 3 3+ (b) f() = (c) f() = (d) f() = (+) /3 (e) f() = sin, [0,π] (g) f() = e (f) f() = sin3, [0,π] (h) f() = (ln)/. At what values of does f have a local maimum or minimum? Sketch the graph. (a) f() = 3 + (b) f() = + (c) f() = e (d) f() = ln (e) f() = (f) f() = 3 (g) f() = sin+cos (h) f() = 3 +3 (i) f() = /3 /3 3. Show that 5 is a critical number of g() = +( 5) 3 but g has no local etreme values at 5.

13 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana Find a cubic function f() = a 3 +b +c+d that has a local maimum value of 3 at and a local minimum value of 0 at. Answer to Eercise 4.4. (a) Inc. on (, ) (, ); dec. on (,) (b) Inc. on (,0) (, ); dec. on (, ) (0,) (c) Inc. on (, ); dec. on (, ) (d) Inc. on (, ); dec. on (, ) (e) Inc. on ( π 3, 5π 3 ) (7π 3,3π); dec. on (0, π 3 ) (5π 3, 7π 3 ) (f) Inc. on (0, π 6 ) (3π 6, 5π 6 ); dec. on (π 6, 3π 6 ) (5π 6,π) (g) Inc. on (0, ); dec. on (,0) (h) Inc. on (0,e ); dec. on (e, ). (a) Loc. ma. at = 7 ; loc. min. at = (b) None 0 (c) Loc. ma. at = 0. (d) None

14 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 96 (e) None (f) Loc. ma. at = 3; Loc. min. at = 3 4 (g) Loc. ma. at = π 4 +nπ; Loc. min. at = 5π 4 +nπ π (h) Loc. ma. at = ; Loc. min. at = 0 (i) Loc. min. at = 4. f() = 9 ( )

15 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana Concavit Definition 4.5. Letf bedifferentiableonanopeninterval I. Wesathatf (aswell asitsgraph) is concave up on I if f is increasing on I and we sa that f is concave down on I if f is decreasing on I. Theorem 4.6 (Concavit Test). Let f be twice differentiable on an open interval I.. If f > 0 for all in I, then f is concave up on I.. If f < 0 for all in I, then f is concave down on I. Definition 4.6. Let f be continuous at c. We call (c,f(c)) an inflection point of the graph f if f is concave up on one side of c and concave down on the other side. Eample 4.5. Let f() = Find the intervals of concavit and the inflection points.

16 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 98 Theorem 4.7 (The Second Derivative Test). Let f and f eist at ever point in an open interval (a,b) containing c.. If f (c) = 0 and f (c) < 0, then f(c) is a local maimum value of f.. If f (c) = 0 and f (c) > 0, then f(c) is a local minimum value of f. 3. If f (c) = 0 and f (c) = 0, then the second derivative test is inconclusive. Eample 4.6. Forf() = 8 (4 8 ), usethesecondderivativetesttoidentiflocaletrema.

17 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 99 Eample 4.7. Let f() = Find the interval of increase and decrease. Find the local maimum and minimum values. Find the intervals of concavit and the inflection points. Use the above information to sketch the graph.

18 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 00 Eample 4.8. Let f() = /3 (6 ) /3. Find the interval of increase and decrease. Find the local maimum and minimum values. Find the intervals of concavit and the inflection points. Use the above information to sketch the graph.

19 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 0 Eercises 4.5. Use the given graph of f to find the intervals of concavit. (a) (b) 4 4 (c) (d) Given the following functions. Find the interval of increase and decrease. Find the local maimum and minimum values. Find the intervals of concavit and the inflection points. Use the above information to sketch the graph. (a) f() = 3 3 (b) f() = 4 6 (c) f() = (d) f() = + (e) f() = /3 (+3) /3 (f) f() = sin

20 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 0 Answer to Eercise 4.5. (a) CU on (, ) (, ); CD on (,) (b) CU on (,0); CD on (0, ) (c) CU on (, ); CD on (,) (d) CU on (,0) (, ); CD on (, ) (0,). (a) Inc. on (, ) (, ); dec. on (,); loc. ma. f( ) = 7; loc. min. f() = 0; CU on (, ); CD on (, ); IP (, 3) (b) Inc. on ( 3,0) ( 3, ); dec. on (, 3) (0, 3); loc. min. f(± 3) = 9; loc. ma. f(0) = 0; CU on (, ) (, ); CD on (,); IP (±, 5) 0 (c) Inc. on (, ) (, ); dec. on (, ); loc. ma. f( ) = 5; loc. min. f() = ; CU on ( /,0) (/, ); CD on (, / ) (/, ); IP (0,3), (±/,3 8 7 ) 5

21 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 03 (d) Inc. on (, ); No loc. ma and loc. min; CU on (0, ); CD on (,0); IP (0,0) e) Inc. on (, 3) (, ); dec. on ( 3, ); loc. ma. f( 3) = 0; loc. min. f( ) = 3 4; CU on (, 3) ( 3,0); CD on (0, ); IP (0,0) (f) Inc. on (0,π/) (π,3π/); dec. on (π/,π) (3π/,π) loc. ma. f(π/) = f(3π/) = ; loc. min. f(π) = 0 CU on (0,π/4) (3π/4,5π/4) (7π/4,π); CD on (π/4,3π/4) (5π/4,7π/4) IP (π/4, ),(3π/4, ),(5π/4, ),(7π/4, ) 0 π π 3π π 4.6 Applied Maimum and Minimum Problems In this section we will show how the methods discussed in the preceding section can be used to solve various applied optimization problems. A Procedure for Solving Applied Maimum and Minimum Problems. Read the problem carefull until it is clearl understood. Ask ourself: What is the unknown? What are the given quantities? What are the given conditions?. Draw an appropriate figure and label the quantities relevant to the problem.

22 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana Find a formula for the quantit to be maimized or minimized. 4. Using the conditions stated in the problem to eliminate variables, epress the quantit to be maimized or minimized as a function of one variable. 5. Find the interval of possible values for this variable from the phsical restrictions in the problem. 6. If applicable, use the techniques of the preceding section to obtain the maimum or minimum. Eample 4.9. A clindrical can is to be made to hold L of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can.

23 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 05 Eample We need to enclose a field with a rectangular fence. We have 500 ft. of fencing material and a building is on one side of the field and so won t need an fencing. Determine the dimensions of the field that will enclose the largest area.

24 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 06 Eample 4.3. Find the radius and height of the right circular clinder of largest volume that can be inscribed in a right circular cone with radius 6 inches and height 0 inches.

25 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 07 Eercises 4.6. Find two positive numbers whose product is 00 and whose sum is minimum.. Findtwo numbers whoseproduct is 6andthesumofwhose squares isminimum. 3. For what number does the principal fourth root eceed twice the number b the largest amount. 4. Find the dimensions of a rectangle with area 000 m whose perimeter is as small as possible. 5. A bo with a square base and open top must have the volume of 3,000 cm 3. Find the dimensions of the bo that minimize the amount of material used. 6. If 00 cm of material is available to make a bo with a square base and an open top, find the largest possible volume of the bo. 7. A farmer wishes to fence off two identical adjoining rectangular pens, each with 900 square feet of area, as shown in the following Figure. What are and so that the least amount of fence is required? 8. Find the point on the line = 4+7 that is closest to the origin. 9. Find the point on the line 6+ = 9 that is closest to the point ( 3,). 0. Find the point on the parabola = that is closest to the point (0,5).. Find the point on the parabola + = 0 that is closest to the point (0, 3).. Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r. 3. Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r. 4. A right circular clinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a clinder. 5. A right circular clinder is inscribed in a sphere of radius r. Find the largest possible surface area of such a clinder. 6. A right circular clinder is inscribed in a sphere of radius r. Find the largest possible volume of such a clinder. 7. Show that the rectangle with maimum perimeter that can be inscribed in a circle is a square. 8. (a) Showthatofalltherectangleswithagivenarea, theonewithsmallest perimeter is a square.

26 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 08 (b) Show that of all the rectangles with a given perimeter, the one with largest area is a square. 9. At 7 : 00am. one ship was 60 miles due east from a second ship. If the first ship sailed west at 0 miles per hour and the second ship sailed southeast at 30 miles per hour, when were the closest together? 0. A piece of wire 0 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maimum? (b) A minimum? Answer to Eercise ,0. 4, ,000 cm ( 8, ) ( 45, ) ( ) ( 0. 3, 9, 3, 9. (, ). r r 3. Base 3r, height 3r/ πr h 5. πr ( + 5 ) 6. 4πr 3 / : 09am. 0. (a) Use all of the wire for the square (b) 40 3/ ( ) m for the square ) 4.7 Rolle s Theorem; Mean Value Theorem In this section we will discuss a result called the Mean Value Theorem. The theorem has so man important consequences that it is regarded as one of the major principle is calculus Theorem 4.8 (Rolle s Theorem). Let f be a function that satisfies the following three hpotheses:. f is continuous on the closed interval [a,b].. f is differentiable on the open interval (a, b). 3. f(a) = f(b) Then there is a number c (a,b) such that f (c) = 0. The following Figure shows the graphs of three such functions. In each case it appears that there is at least one point ( c,f(c) ) on the graph where the tangent is horizontal and therefore f (c) = 0.

27 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 09 a c b a c b a c c b Eample 4.3. Verif that the function f() = +3, [ 3,0] satisfies the three hpotheses of Rolle s Theorem on the given interval. Then find all numbers c that satisf the conclusion of Rolle s Theorem.

28 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana 0 Theorem 4.9 (Mean Value Theorem). Let f be a function that satisfies the following hpotheses:. f is continuous on the closed interval [a,b].. f is differentiable on the open interval (a, b). Then there is a number c (a,b) such that f (c) = f(b) f(a) b a or, equivalentl, f(b) f(a) = f (c)(b a) Eample Let f() = 3 + 4, [,] Find all numbers c that satisf the conclusion of the Mean Value Theorem.

29 MA: Prepared b Asst.Prof.Dr. Archara Pacheenburawana Eercises 4.7. Verif that the function satisfies the three hpotheses of Rolle s Theorem on the given interval. Then find all numbers c that satisf the conclusion of Rolle s Theorem. (a) f() = 4+, [0,4] (b) f() = , [0,] (c) f() = sinπ, [,] (d) f() = +6, [ 6,0]. Use the graph of f to estimate the values of c that satisf the conclusion of the Mean Value Theorem for the interval [0, 8]. = f() 3. Verif that the function satisfies the hpotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisf the conclusion of the Mean Value Theorem. (a) f() = 3 ++5, [,] (b) f() = 3, [0,] (c) f() = e, [0,3] (d) f() = +, [,4] Answer to Eercise 4.7. (a) (b) 3± 3 3 (c) ± 4,±3 4 (d) , 3., 4.4, (a) 0 (b) 3 (c) ln[( e 6 )/6] (d) 3

Mth Test 3 Review Stewart 8e Chapter 4. For Test #3 study these problems, the examples in your notes, and the homework.

Mth Test 3 Review Stewart 8e Chapter 4. For Test #3 study these problems, the examples in your notes, and the homework. For Test #3 study these problems, the eamples in your notes, and the homework. I. Absolute Etrema A function, continuous on a closed interval, always has an absolute maimum and absolute minimum. They occur