Optimal locations Suppose n houses are located at the

Cardboard boxes A lidless cardboard box is to be made with a volume of 4 \(m^{3}\). Find the dimensions of the box that requires the least amount of cardboard.

Describe the appearance of a smooth surface with a local maximum at a point

Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given set R. \(f(x, y)=-x^{2}-y^{2}+\sqrt{3} \); \(R=\left\{(x, y): x^{2}+y^{2} \leq 6\right\}\)

Explain how the Second Derivative Test is used

Cardboard boxes A lidless box is to be made using 2 \(m^{2}\) of cardboard. Find the dimensions of the box with the largest possible volume.

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=x^{4}+2 y^{2}-4 x y\)

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=2 x y e^{-x^{2}-y^{2}}\)

Optimal box Find the dimensions of the rectangular box with maximum volume in the first octant with one vertex at the origin and the opposite vertex on the ellipsoid \(36 x^{2}+4 y^{2}+9 z^{2}=36\)

Second Derivative Test Prove that if (a, b) is a critical point off at which \(f_{x}(a, b)=f_{y}(a, b)=0\) and \(f_{x x}(a, b)<0<f_{y y}(a, b)\) or \(f_{y y}(a, b)<0<f_{x x}(a, b)\), then f has a saddle point at (a, b).

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=\frac{x-y}{1+x^{2}+y^{2}}\)

If \(f_{x}(a, b)=f_{y}(a, b)=0\), does it follow that f has a local maximum or local minimum at (a,b)? Explain.

Extreme points from contour plots Based on the level curves that are visible in the following graphs, identify the approximate locations of the local maxima, local minima, and saddle points.

Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given set R. \(f(x, y)=x^{2}+y^{2}-2 x-2 y\); R is the closed set bounded by the triangle with vertices (0,0), (2,0), and (0,2)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume that f is differentiable at the points in question. a. The fact that \(f_{x}(2,2)=f_{y}(2,2)=0\) implies that f has a local maximum, local minimum, or saddle point at (2, 2). b. The function f could have a local maximum at (a, b) where \(f_{y}(a, b) \neq 0\) c. The function f could have both an absolute maximum and an absolute minimum at two different points that are not critical points. d. The tangent plane is horizontal at a point on a surface corresponding to a critical point.

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=\tan ^{-1}(x y)\)

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=\sqrt{x^{2}+y^{2}-4 x+5}\)

Critical points Find all critical points of the following functions \(f(x, y)=3 x^{2}-4 y^{2}\)

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=x y e^{-x-y}\)

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=\frac{x y(x-y)}{x^{2}+y^{2}}\)

Maximum area triangle Among all triangles with a perimeter of 9 units, find the dimensions of the triangle with the maximum area. It may be easiest to use Heron's formula, which states that the area of a triangle with side length a, b, and c is \(A=\sqrt{s}(s-a)(s-b)(s-c)\), where 2s is the perimeter of the triangle.

What are the conditions for a critical point of a function f?

Absolute extrema on open and/or unbounded sets Find the point on the plane x + y + z = 4 nearest the point P(0,3,6).

Absolute extrema on open and/or unbounded sets If possible, find the absolute maximum and minimum values of the following functions on the set R. \(f(x, y)=2 e^{-x-y} ; R=\{(x, y): x \geq 0, y \geq 0\}\)

Extreme points from contour plots Based on the level curves that are visible in the following graphs, identify the approximate locations of the local maxima, local minima, and saddle points.

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=x^{2}-x^{4} / 2-y^{2}-x y\)

Least distance What point on the plane x - y + z = 2 is closest to the point (1, 1, 1)?

What is an absolute minimum value of a function f on a set R in \(R^{2}\)?

Critical points Find all critical points of the following functions \(f(x, y)=x^{4}+y^{4}-16 x y\)

Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given set R. \(f(x, y)=4+2 x^{2}+y^{2} \\\) \(R=\{(x, y):-1 \leq x \leq 1,-1 \leq y \leq 1\}\)

What is the discriminant and how do you compute it?

Shipping regulations A shipping company handles rectangular boxes provided the sum of the height and the girth of the box does not exceed 96 in. (The girth is the perimeter of the smallest base of the box.) Find the dimensions of the box that meets this condition and has the largest volume.

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=x^{4}+y^{4}-4 x-32 y+10\)

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=-4 x^{2}+8 y^{2}-3\)

Two mountains without a saddle Show that the following two functions have two local maxima but no other extreme points (thus no saddle or basin between the mountains). a. \(f(x, y)=-\left(x^{2}-1\right)^{2}-\left(x^{2}-e^{y}\right)^{2}\) b. \(f(x, y)=4 x^{2} e^{y}-2 x^{4}-e^{4 y}\) Source: Proposed by Ira Rosenholtz, Mathematics Magazine (February, 1987).

Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given set R. \(f(x, y)=x^{2}+y^{2}+4 x-2 y ; R=\left\{(x, y): x^{2}+y^{2} \leq 16\right\}\)

Critical points Find all critical points of the following functions \(f(x, y)=(3 x-2)^{2}+(y-4)^{2}\)

Inconclusive tests Show that the Second Derivative test is inconclusive when applied to the following functions at (0,0). Describe the behavior of the function at the critical point. \(f(x, y)=\sin \left(x^{2} y^{2}\right)\)

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=(4 x-1)^{2}+(2 y+4)^{2}+1\)

Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given set R. \(f(x, y)=x^{2}+y^{2}-2 y+1 ; R=\left\{(x, y): x^{2}+y^{2} \leq 4\right\}\)

Critical points Find all critical points of the following functions \(f(x, y)=x^{3} / 3-y^{3} / 3+3 x y\)

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=4+2 x^{2}+3 y^{2}\)

Inconclusive tests Show that the Second Derivative test is inconclusive when applied to the following functions at (0,0). Describe the behavior of the function at the critical point. \(f(x, y)=x^{4} y^{2}\)

Least squares approximation In its many guises, least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\). The data may be plotted as a scatterplot in the xy-plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Least squares practice Use the results of Exercise 63 to find the best-fit line for following data sets. Plot the points and the best-fit line. (-1,0), (0, 6), (3,8)

Describe the usual appearance of a smooth surface at a saddle point

What is the procedure for locating absolute maximum and minimum values on a closed bounded domain?

Critical points Find all critical points of the following functions \(f(x, y)=1+x^{2}+y^{2}\)

Critical points Find all critical points of the following functions \(f(x, y)=x^{2}-6 x+y^{2}+8 y\)

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=y e^{x}-e^{y}\)

Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility \(f(x, y)=\sin (2 \pi x) \cos (\pi y)\) for \(|x| \leq \frac{1}{2} \text { and }|y| \leq \frac{1}{2}\)

Optimal box Find the dimensions of the largest rectangular box in the first octant of the xyz-coordinate system that has one vertex at the origin and the opposite vertex on the plane x + 2y + 3z = 6.

Inconclusive tests Show that the Second Derivative test is inconclusive when applied to the following functions at (0,0). Describe the behavior of the function at the critical point. \(f(x, y)=4+x^{4}+3 y^{4})

Inconclusive tests Show that the Second Derivative test is inconclusive when applied to the following functions at (0,0). Describe the behavior of the function at the critical point. \(f(x, y)=x^{2} y-3)

Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given set R. \(f(x, y)=6-x^{2}-4 y^{2} \\\); \(R=\{(x, y):-2 \leq x \leq 2,-1 \leq y \leq 1\}\)

Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given set R. \(f(x, y)=x^{2}+4 y^{2}+2 x+4 y\); R is the closed set bounded by the ellipse \(\{(x, y): x=4 \cos \theta, y=\sin \theta, \text { for } 0 \leq \theta \leq 2 \pi\} \text {. }\)

Absolute maxima and minima Find the absolute maximum and minimum values of the following functions on the given set R. \(f(x, y)=\sqrt{x^{2}+y^{2}-2 x+2}\); R is the closed half disk \(\left\{(x, y): x^{2}+y^{2} \leq 4 \text { with } y \geq 0\right\}\)

Absolute extrema on open and/or unbounded sets If possible, find the absolute maximum and minimum values of the following functions on the set R. \(f(x, y)=x^{2}+y^{2}-4 ; R=\left\{(x, y): x^{2}+y^{2}<4\right\}\)

Absolute extrema on open and/or unbounded sets If possible, find the absolute maximum and minimum values of the following functions on the set R. \(f(x, y)=x+3 y ; R=\{(x, y):|x|<1,|y|<2\}\)

Absolute extrema on open and/or unbounded sets If possible, find the absolute maximum and minimum values of the following functions on the set R. \(f(x, y)=x^{2}-y^{2} ; R=\{(x, y):|x|<1,|y|<1\}\)

Absolute extrema on open and/or unbounded sets Find the point(s) on the cone \(z^{2}=x^{2}+y^{2}\) nearest the point P(1,4,0).

Absolute extrema on open and/or unbounded sets Find the point on the surface \(f(x, y)=x^{2}+y^{2}+10\) nearest the plane x + 2y - z = 0. Identify the point on the plane.

Absolute extrema on open and/or unbounded sets Rectangular boxes with a volume of 10 \(m^{3}\) are to be made of two materials. The material for the top and bottom of the box costs $8/\(m^{2}\) and the material for the sides of the box costs SI/\(m^{2}\). What are the dimensions of the box that minimizes the cost of the box?

Maximum/Minimum of linear functions Let R be a closed bounded set in R and let f(x, y) = ax + by + c, where a, b, and c are real numbers, with a and b not both zero. Give a geometrical argument explaining why the absolute maximum and minimum values of f over R occur on the boundaries of \(R^{2}\).

Magic triples Let x, y, and z be nonnegative numbers with x + y + z = 200. a. Find the values of x, y, and z that minimize \(x^{2}+y^{2}+z^{2}\)? b. Find the values of x, y, and that minimize \(\sqrt{x^{2}+y^{2}+z^{2}}\)? c. Find the values of x, y, and that maximize xyz. d. Find the values of x, y, and z that maximize \(x^{2} y^{2} z^{2}\)

Powers and roots Assume that x + y + z = 1 with x \(\geq\) 0, y \(\geq\)0, and z \(\geq\) 0. a. Find the maximum and minimum values of \(\left(1+x^{2}\right)\left(1+y^{2}\right)\left(1+z^{2}\right)\) b. Find the maximum and minimum values of \((1+\sqrt{x})(1+\sqrt{y})(1+\sqrt{z})\) (Source: Math Horizons (April 2004).]

Optimal locations Suppose n houses are located at the distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\). A power substation must be located at a point such that the sum of the squares of the distances between the houses and the substation is minimized. a. Find the optimal location of the substation in the case that n = 3 and the houses are located at (0,0), (2,0), and (1,1). b. Find the optimal location of the substation in the case that n-3 and the houses are located at distinct points \(\left(x_{1}, y_{1}\right)\), \(\left(x_{2}, y_{2}\right)\), and \(\left(x_{3}, y_{3}\right)\). c. Find the optimal location of the substation in the general case of n houses located at distinct points \(\left(x_{1}, y_{1}\right)\), \(\left(x_{1}, y_{2}\right)\),..., \(\left(x_{n}, y_{n}\right)\) d. You might argue that the locations found in parts (a), (b) and (c) are not optimal because they result from minimizing the sum of the squares of the distances, not the sum of the distances themselves. Use the locations in part (a) and write the function that gives the sum of the distances. Note that minimizing this function is much more difficult than in part (a). Then use a graphing utility to determine whether the optimal location is the same in the two cases. (Also see Exercise 69 about Steiner's problem.)

Least squares approximation In its many guises, least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\). The data may be plotted as a scatterplot in the xy-plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Let the equation of the best-fit line be y = mx + b, where the slope m and the y-intercept b must be determined using the least squares condition. First assume that there are three data points (1,2), (3,5), and (4, 6). Show that the function of m and b that gives the sum of the squares of the vertical distances between the line and the three data points is \(E(m, b)=& {[(m+b)-2]^{2}+[(3 m+b)-5]^{2} } +[(4 m+b)-6]^{2}\) Find the critical points of E and find the values of m and b that minimize E. Graph the three data points and the best-fit line.

Least squares approximation In its many guises, least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\). The data may be plotted as a scatterplot in the xy-plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Generalize the procedure in Exercise 62 by assuming that n data points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) are given. Write the function Em, b) (summation notation allows for a more compact calculation). Show that the coefficients of the best-fit line are \(m=\frac{\left(\sum x_{k}\right)\left(\sum y_{k}\right)-n \sum x_{k} y_{k}}{\left(\sum x_{k}\right)^{2}-n \sum x_{k}^{2}}\) \(b=\frac{1}{n}\left(\sum y_{k}-m \sum x_{k}\right)\) where all sums run from k = 1 to k = n.

Least squares approximation In its many guises, least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\). The data may be plotted as a scatterplot in the xy-plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Least squares practice Use the results of Exercise 63 to find the best-fit line for following data sets. Plot the points and the best-fit line. (0,0), (2, 3), (4,5)

Ellipsoid inside a tetrahedron (1946 Putnam Exam) Let P be a plane tangent to the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1\) at a point in the first octant. Let T be the tetrahedron in the first octant bounded by P and the coordinate planes x = 0, y = 0, and z = 0. Find the minimum volume of T. (The volume of a tetrahedron is one-third the area of the base times the height.)

Steiner's problem for three points Given three distinct noncollinear points A, B, and C in the plane, find the point in the plane such that the sum of the distances |AP| + |BP| + |CP| is a minimum. Here is how to proceed with three points, assuming that the triangle formed by the three points has no angle greater than \(2 \pi / 3\left(120^{\circ}\right)\). a. Assume the coordinates of the three given points are A(\(x_{1}, y_{1}\)), B(\(x_{2}, y_{2}\)), and C(\(x_{3}, y_{3}\)). Let \(d_{1}\)(x, y) be the distance between A(\(x_{1}, y_{1}\)) and a variable point P(x, y). Compute the gradient of \(d_{1}\) and show that it is a unit vector pointing along the line between the two points. b. Define \(d_{2}\) and \(d_{3}\) in a similar way and show that \(\nabla d_{2}\), and \(\nabla d_{3}\) are also unit vectors in the direction of the line between the two points. c. The goal is to minimize \(f(x, y)=d_{1}+d_{2}+d_{3}\). Show that the condition \(f_{x}=f_{y}=0\) implies that \(\nabla d_{1}+\nabla d_{2}+\nabla d_{3}=0\) d. Explain why part (c) implies that the optimal point P has the property that the three line segments AP, BP, and CP all intersect symmetrically in angles of 2\(\pi\)/3. e. What is the optimal solution if one of the angles in the triangle is greater than 2\(\pi\)/3 (just draw a picture)? f. Estimate the Steiner point for the three points (0,0), (0, 1), (2,0).

Slicing plane Find an equation of the plane passing through the point (3, 2, 1) that slices off the region in the first octant with the least volume.

Solitary critical points A function of one variable has the property that a local maximum (or minimum) occurring at the only critical point is also the absolute maximum (or minimum) (for example, \(\left.f(x)=x^{2}\right)\). Does the same result hold for a function of two variables? Show that the following functions have the property that they have a single local maximum (or minimum), occurring at the only critical point, but that the local maximum (or minimum) is not an absolute maximum (or minimum) on R. a. \(f(x, y)=3 x e^{y}-x^{3}-e^{3 y}\) b. \(f(x, y)=\left(2 y^{2}-y^{4}\right)\left(e^{x}+\frac{1}{1+x^{2}}\right)-\frac{1}{1+x^{2}}\) This property has the following interpretation. Suppose that a surface has a single local minimum that is not the absolute minimum. Then water can be poured into the basin around the local minimum and the surface never overflows, even though there are points on the surface below the local minimum. Source: See three articles in Mathematics Magazine (May 1985) and Calculus and Analytical Geometry, 2nd ed., Philip Gillett.