D u f f x h f y k. Applying this theorem a second time, we have. f xx h f yx k h f xy h f yy k k. f xx h 2 2 f xy hk f yy k 2

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1 93 CHAPTER 4 PARTIAL DERIVATIVES We close this section b giving a proof of the first part of the Second Derivatives Test. Part (b) has a similar proof. PROOF OF THEOREM 3, PART (A) We compute the second-order directional derivative of f in the direction of u h, k. The first-order derivative is given b Theorem 4.6.3: D u f f h f k Appling this theorem a second time, we have D u f D u D u f D u f h D u f k f h f k h f h f k k f h f hk f k (b Clairaut s Theorem) If we complete the square in this epression, we obtain D u f f h f f k k f f f f We are given that f a, b and D a, b. But f and D f f f are continuous functions, so there is a disk B with center a, b and radius such that f, and D, whenever, is in B. Therefore, b looking at Equation, we see that D u f, whenever, is in B. This means that if C is the curve obtained b intersecting the graph of f with the vertical plane through P a, b, f a, b in the direction of u, then C is concave upward on an interval of length. This is true in the direction of ever vector u, so if we restrict, to lie in B, the graph of f lies above its horizontal tangent plane at P. Thus f, f a, b whenever, is in B. This shows that f a, b is a local minimum. M 4.7 EXERCISES. Suppose, is a critical point of a function f with continuous second derivatives. In each case, what can ou sa about f? (a) f, 4, f,, f, (b) f, 4,. Suppose (, ) is a critical point of a function t with continuous second derivatives. In each case, what can ou sa about t? (a) t,, t, 6, t, (b) t,, (c) t, 4, f, 3, t,, t, 6, f, t, 8 t, Use the level curves in the figure to predict the location of the critical points of f and whether f has a saddle point or a local maimum or minimum at each critical point. Eplain our reasoning. Then use the Second Derivatives Test to confirm our predictions. 3. f, _ _

2 SECTIO 4.7 MAXIMUM AD MIIMUM VALUES f, _.9 _.7 _.5.5 _ f, e 3. f, sin sin sin,, 4. f, sin sin cos, 4, 4 ; 5 8 Use a graphing device as in Eample 4 (or ewton s method or a rootfinder) to find the critical points of f correct to three decimal places. Then classif the critical points and find the highest or lowest points on the graph. 5. f, Find the local maimum and minimum values and saddle point(s) of the function. If ou have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 5. f, f, f, f, e 4 9. f,. f, 3 5. f, f, 3. f, e cos 4. f, cos 5. f, e 6. f, e 7. f, cos, 8. f, sin sin,, 9. Show that f, 4 4 has an infinite number of critical points and that D at each one. Then show that f has a local (and absolute) minimum at each critical point.. Show that f has maimum values at (, s ), e and minimum values at (, s ). Show also that f has infinitel man other critical points and D at each of them. Which of them give rise to maimum values? Minimum values? Saddle points? ; 4 Use a graph and/or level curves to estimate the local maimum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisel.. f, f, f, f, e 4 3 4cos 9 36 Find the absolute maimum and minimum values of f on the set D. 9. f, 4 5, D is the closed triangular region with vertices,,,, and, 3 3. f, 3, D is the closed triangular region with vertices,, 5,, and, 4 3. f, 4, D,, 3. f, 4 6, D, 4, f, 4, D, 4 4 3, 34. f,, 35. f, 3 4, D,,, 3 D, 36. f, 3 3 3, D is the quadrilateral whose vertices are, 3,, 3,,, and,. ; 37. For functions of one variable it is impossible for a continuous function to have two local maima and no local minimum. But for functions of two variables such functions eist. Show that the function f, has onl two critical points, but has local maima at both of them. Then use a computer to produce a graph with a carefull chosen domain and viewpoint to see how this is possible. ; 38. If a function of one variable is continuous on an interval and has onl one critical number, then a local maimum has to be

3 93 CHAPTER 4 PARTIAL DERIVATIVES an absolute maimum. But this is not true for functions of two variables. Show that the function has eactl one critical point, and that f has a local maimum there that is not an absolute maimum. Then use a computer to produce a graph with a carefull chosen domain and viewpoint to see how this is possible. 39. Find the shortest distance from the point,, to the plane z. 4. Find the point on the plane z 4 that is closest to the point,, Find the points on the cone z that are closest to the point 4,,. 4. Find the points on the surface 9 z that are closest to the origin. 43. f, 3e 3 e 3 Find three positive numbers whose sum is and whose product is a maimum. 44. Find three positive numbers whose sum is and the sum of whose squares is as small as possible. 45. Find the maimum volume of a rectangular bo that is inscribed in a sphere of radius r. 46. Find the dimensions of the bo with volume cm 3 that has minimal surface area. 47. Find the volume of the largest rectangular bo in the first octant with three faces in the coordinate planes and one verte in the plane 3z Find the dimensions of the rectangular bo with largest volume if the total surface area is given as 64 cm. 49. Find the dimensions of a rectangular bo of maimum volume such that the sum of the lengths of its edges is a constant c. 5. The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials. 5. A cardboard bo without a lid is to have a volume of 3, cm 3. Find the dimensions that minimize the amount of cardboard used. 5. A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a rate of units m per da, the north and south walls at a rate of 8 units m per da, the floor at a rate of unit m per da, and the roof at a rate of 5 units m per da. Each wall must be at least 3 m long, the height must be at least 4 m, and the volume must be eactl 4 m 3. (a) Find and sketch the domain of the heat loss as a function of the lengths of the sides. (b) Find the dimensions that minimize heat loss. (Check both the critical points and the points on the boundar of the domain.) (c) Could ou design a building with even less heat loss if the restrictions on the lengths of the walls were removed? 53. If the length of the diagonal of a rectangular bo must be L, what is the largest possible volume? 54. Three alleles (alternative versions of a gene) A, B, and O determine the four blood tpes A (AA or AO), B (BB or BO), O (OO), and AB. The Hard-Weinberg Law states that the proportion of individuals in a population who carr two different alleles is where p, q, and r are the proportions of A, B, and O in the population. Use the fact that p q r to show that P is at most Suppose that a scientist has reason to believe that two quantities and are related linearl, that is, m b, at least approimatel, for some values of m and b. The scientist performs an eperiment and collects data in the form of points,,,,..., n, n, and then plots these points. The points don t lie eactl on a straight line, so the scientist wants to find constants m and b so that the line m b fits the points as well as possible. (See the figure.) (, ) Let d i i m i b be the vertical deviation of the point i, i from the line. The method of least squares determines and so as to minimize n m b i d i, the sum of the squares of these deviations. Show that, according to this method, the line of best fit is obtained when m n P pq pr rq i m n i m i +b i b n i bn n Thus the line is found b solving these two equations in the two unknowns m and b. (See Section. for a further discussion and applications of the method of least squares.) 56. Find an equation of the plane that passes through the point,, 3 and cuts off the smallest volume in the first octant. i d i ( i, i ) i n i i i i i

4 DISCOVERY PROJECT QUADRATIC APPROXIMATIOS AD CRITICAL POITS 933 APPLIED PROJECT DESIGIG A DUMPSTER For this project we locate a trash dumpster in order to stud its shape and construction. We then attempt to determine the dimensions of a container of similar design that minimize construction cost.. First locate a trash dumpster in our area. Carefull stud and describe all details of its construction, and determine its volume. Include a sketch of the container.. While maintaining the general shape and method of construction, determine the dimensions such a container of the same volume should have in order to minimize the cost of construction. Use the following assumptions in our analsis: The sides, back, and front are to be made from -gauge (.46 inch thick) steel sheets, which cost $.7 per square foot (including an required cuts or bends). The base is to be made from a -gauge (.345 inch thick) steel sheet, which costs $.9 per square foot. Lids cost approimatel $5. each, regardless of dimensions. Welding costs approimatel $.8 per foot for material and labor combined. Give justification of an further assumptions or simplifications made of the details of construction. 3. Describe how an of our assumptions or simplifications ma affect the final result. 4. If ou were hired as a consultant on this investigation, what would our conclusions be? Would ou recommend altering the design of the dumpster? If so, describe the savings that would result. DISCOVERY PROJECT QUADRATIC APPROXIMATIOS AD CRITICAL POITS The Talor polnomial approimation to functions of one variable that we discussed in Chapter can be etended to functions of two or more variables. Here we investigate quadratic approimations to functions of two variables and use them to give insight into the Second Derivatives Test for classifing critical points. In Section 4.4 we discussed the linearization of a function f of two variables at a point a, b : L, f a, b f a, b a f a, b b Recall that the graph of L is the tangent plane to the surface z f, at a, b, f a, b and the corresponding linear approimation is f, L,. The linearization L is also called the first-degree Talor polnomial of f at a, b.. If f has continuous second-order partial derivatives at a, b, then the second-degree Talor polnomial of f at a, b is Q, f a, b f a, b a f a, b b f a, b a f a, b a b f a, b b and the approimation f, Q, is called the quadratic approimation to f at a, b. Verif that Q has the same first- and second-order partial derivatives as f at a, b.

5 934 CHAPTER 4 PARTIAL DERIVATIVES. (a) Find the first- and second-degree Talor polnomials and of f, e L Q at (, ). ; (b) Graph f, L, and Q. Comment on how well L and Q approimate f. 3. (a) Find the first- and second-degree Talor polnomials L and Q for f, e at (, ). (b) Compare the values of L, Q, and f at (.9,.). ; (c) Graph f, L, and Q. Comment on how well L and Q approimate f. 4. In this problem we analze the behavior of the polnomial f, a b c (without using the Second Derivatives Test) b identifing the graph as a paraboloid. (a) B completing the square, show that if a, then f, a b c a b a 4ac b 4a (b) Let D 4ac b. Show that if D and a, then f has a local minimum at (, ). (c) Show that if D and a, then f has a local maimum at (, ). (d) Show that if D, then (, ) is a saddle point. 5. (a) Suppose f is an function with continuous second-order partial derivatives such that f, and (, ) is a critical point of f. Write an epression for the second-degree Talor polnomial, Q, of f at (, ). (b) What can ou conclude about Q from Problem 4? (c) In view of the quadratic approimation f, Q,, what does part (b) suggest about f? g(, )=k FIGURE 4.8 f(, )= f(, )= f(, )=9 f(, )=8 f(, )=7 TEC Visual 4.8 animates Figure for both level curves and level surfaces. LAGRAGE MULTIPLIERS In Eample 6 in Section 4.7 we maimized a volume function V z subject to the constraint z z, which epressed the side condition that the surface area was m. In this section we present Lagrange s method for maimizing or minimizing a general function f,, z subject to a constraint (or side condition) of the form t,, z k. It s easier to eplain the geometric basis of Lagrange s method for functions of two variables. So we start b tring to find the etreme values of f, subject to a constraint of the form t, k. In other words, we seek the etreme values of f, when the point, is restricted to lie on the level curve t, k. Figure shows this curve together with several level curves of f. These have the equations f, c, where c 7, 8, 9,,. To maimize f, subject to t, k is to find the largest value of c such that the level curve f, c intersects t, k. It appears from Figure that this happens when these curves just touch each other, that is, when the have a common tangent line. (Otherwise, the value of c could be increased further.) This means that the normal lines at the point, where the touch are identical. So the gradient vectors are parallel; that is, f, for some scalar. This kind of argument also applies to the problem of finding the etreme values of f,, z subject to the constraint t,, z k. Thus the point,, z is restricted to lie on the level surface S with equation t,, z k. Instead of the level curves in Figure, we consider the level surfaces f,, z c and argue that if the maimum value of f is f,, z c, then the level surface f,, z c is tangent to the level surface t,, z k and so the corresponding gradient vectors are parallel. t,

f x, y x 2 y 2 2x 6y 14. Then

f x, y x 2 y 2 2x 6y 14. Then SECTION 11.7 MAXIMUM AND MINIMUM VALUES 645 absolute minimum FIGURE 1 local maimum local minimum absolute maimum Look at the hills and valles in the graph of f shown in Figure 1. There are two points a,