Solved: ?Find the local maximum and minimum values and saddle point(s) of the function

Suppose (1, 1) is a critical point of a function f with continuous second derivatives. In each case, what can you say about f? (a) fxx(1, 1) = 4, fxy(1, 1) = 1, fyy(1, 1) = 2 (b) fxx(1, 1) = 4, fxy(1, 1) = 3, fyy(1, 1) = 2

Suppose (0, 2) is a critical point of a function g with continuous second derivatives. In each case, what can you say about g? (a) gxx(0, 2) = ? 1, gxy(0, 2) = 6, gyy(0, 2) = 1 (b) gxx(0, 2) = ? 1, gxy(0, 2) = 2, gyy(0, 2) = ? 8 (c) gxx(0, 2) = 4, gxy(0, 2) = 6, gyy(0, 2) = 9

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 maximum or minimum at each critical point. Explain your reasoning. Then use the Second Derivatives Test to confirm your predictions. f(x, y) = 4 + x3 + y3 ? 3xy

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 maximum or minimum at each critical point. Explain your reasoning. Then use the Second Derivatives Test to confirm your predictions. f(x, y) = 3x ? x3 ? 2y2 + y 4

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. f(x, y) = x2 + xy + y2 + y

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. f(x, y) = xy ? 2x ? 2y ? x2 ? y2

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. f(x, y) = 2x2 ? 8xy + y 4 ? 4y3

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. f(x, y) = x3 + y3 + 3xy

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. f(x, y) = (x ? y) (1 ? xy)

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. \(f(x, \ y)=y\left(e^{x}-1\right)\)

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. f(x, y) =

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. f(x, y) = 2 ? x4 + 2x2 ? y 2

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. \(f(x, y)=x^3-3 x-3 x y^2\)

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. \(f(x, y)=x^3+y^3-3 x^2-3 y^2-9 x\)

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. f(x, y) = x4 ? 2x2 + y3 ? 3y

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. f(x, y) = x2 + y4 + 2xy

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. f(x, y) = xy ? x2y ? xy2

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. f(x, y) = (6x ? x2)(4y ? y2)

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. f(x, y) = ex cos y

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. \(f(x, y)=\left(x^{2}+y^{2}\right) e^{-x}\) Equation Transcription: Text Transcription: f(x,y)=(x^2+y^2)e^-x

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. \(f(x, \ y)=y^{2}-2 y \cos \ x, \quad-1 \leqslant x \leqslant 7\)

Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all the important aspects of the function. \(f(x, \ y)=\sin \ x \sin \ y, \quad-\pi<x<\pi, \quad-\pi<y<\pi\)

Show that \(f(x, y)=x^{2}+4 y^{2}-4 x y+2\) has an infinite number of critical points and that \(D=0\) at each one. Then show that f has a local (and absolute) minimum at each critical point. Equation Transcription: Text Transcription: f(x,y)=x^2+4y^2-4xy+2 D=0

Show that \(f(x, y)=x^{2} y e^{-x^{2}-y^{2}}\) has maximum values at \((\pm 1,1 / \sqrt{2})\) and minimum values at \((\pm 1,-1 / \sqrt{2})\). Show also that f has infinitely many other critical points and \(D=0\) at each of them. Which of them give rise to maximum values? Minimum values? Saddle points? Equation Transcription: Text Transcription: f(x,y)=x^2ye^-x^2 -y^2 (1,1/2) (1,-1/2) f D=0

Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. \(f(x, y)=x^{2}+y^{2}+x^{-2} y^{-2}\) Equation Transcription: Text Transcription: f(x,y)=x^2+y^2+x^-2 y^-2

Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. \(f(x, y)=(x-y) e^{-x^{2}-y^{2}}\) Equation Transcription: Text Transcription: f(x,y)=(x-y)e^-x^2 -y^2

Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. \(f(x, y)=\sin x+\sin y+\sin (x+y), \quad 0 \leqslant x \leqslant 2 \pi, \quad 0 \leqslant y \leqslant 2 \pi\)

Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. \(f(x, y)=\sin x+\sin y+\cos (x+y), \quad 0 \leqslant x \leqslant \pi / 4, \quad 0 \leqslant y \leqslant \pi / 4\)

Find the critical points of \(f\) correct to three decimal places (as in Example \(4\)). Then classify the critical points and find the highest or lowest points on the graph, if any. \(f(x, y)=x^{4}+y^{4}-4 x^{2} y+2 y\) Equation Transcription: Text Transcription: f 4 f(x,y)=x^4+y^4-4x^2y+2y

Find the critical points of \(f\) correct to three decimal places (as in Example \(4\)). Then classify the critical points and find the highest or lowest points on the graph, if any. \(f(x, y)=y^{6}-2 y^{4}+x^{2}-y^{2}+y\) Equation Transcription: Text Transcription: f 4 f(x,y)=y^6-2y^4+x^2+y

Find the critical points of \(f\) correct to three decimal places (as in Example \(4\)). Then classify the critical points and find the highest or lowest points on the graph, if any. \(f(x, y)=x^{4}+y^{3}-3 x^{2}+y^{2}+x-2 y+1\) Equation Transcription: Text Transcription: f 4 f(x,y)=x^4+y^3-3x^2+y^2+x-2y+1

Find the critical points of \(f\) correct to three decimal places (as in Example \(4\)). Then classify the critical points and find the highest or lowest points on the graph, if any. \(f(x, y)=20 e^{-x^{2}-y^{2}} \sin 3 x \cos 3 y, \quad|x| \leqslant 1, \quad|y| \leqslant 1\) Equation Transcription: Text Transcription: f 4 f(x,y)=20e^-x^2 -y^2 sin 3x cos 3y, |x| leqslant 1, |y| leqslant 1

Find the absolute maximum and minimum values of f on the set D. \(f(x, y)=x^2+y^2-2 x\), D is the closed triangular region with vertices, (2, 0), (0, 2), and (0, -2)

Find the absolute maximum and minimum values of \(f\) on the set \(D\). \(f(x, y)=x+y-x y\), \(D\) is the closed triangular region with vertices, \((0,0),(0,2)\), and \((4,0)\)

Find the absolute maximum and minimum values of f on the set D. \(f(x, y)=x^{2}+y^{2}+x^{2} y+4, \quad D=\{(x, y) \| x|\leqslant 1, \quad| y \mid \leqslant 1\}\)

Find the absolute maximum and minimum values of \(f\) on the set \(D\). \(f(x, y)=x^{2}+x y+y^{2}-6 y, \quad D=\{(x, y) \mid-3 \leqslant x \leqslant 3, \quad 0 \leqslant y \leqslant 5\}\) Equation Transcription: Text Transcription: f D f(x,y)=x^2+xy+y^2-6y, D={(x,y) | -3 leqslant x leqslant 3, 0 leqslant y leqslant 5}

Find the absolute maximum and minimum values of \(f\) on the set \(D\). \(f(x, y)=x^{2}+2 y^{2}-2 x-4 y+1, \quad D=\{(x, y) \mid 0 \leqslant x \leqslant 2, \quad 0 \leqslant y \leqslant 3\}\) Equation Transcription: Text Transcription: f D f(x,y)=x^2+2y^2-2x-4y+1, D={(x,y) | 0 leqslant x leqslant 2, 0 leqslant y leqslant 3}

Find the absolute maximum and minimum values of \(f\) on the set \(D\). \(f(x, y)=x y^{2}, \quad D=\left\{(x, y) \mid x \geqslant 0, \quad y \geqslant 0, x^{2}+y^{2} \leqslant 3\right\}\) Equation Transcription: Text Transcription: f D f(x,y)=xy^2, D={(x,y) | x geqslant 0, y geqslant 0, x^2+y^2 leqslant 3}

Find the absolute maximum and minimum values of \(f\) on the set \(D\). \(f(x, y)=2 x^{3}+y^{4}, \quad D=\left\{(x, y) \mid x^{2}+y^{2} \leqslant 1\right\}\) Equation Transcription: Text Transcription: f D f(x,y)=2x^3 + y^4, D={(x,y) | x^2+y^2 leqslant 1}

Find the absolute maximum and minimum values of \(f\) on the set \(D\). \(f(x, y)=x^{3}-3 x-y^{3}-12 y\), \(D\) is the quadrilateral whose vertices are \((?2, 3), (2, 3), (2, 2)\), and \((?2, ?2)\) Equation Transcription: Text Transcription: f D f(x,y)=x^3-3x-y^3-12y (?2, 3), (2, 3), (2, 2), and (?2, ?2)

For functions of one variable it is impossible for a continuous function to have two local maxima and no local minimum. But for functions of two variables such functions exist. Show that the function \(f\left(x_{r} y\right)=-\left(x^{2}-1\right)^{2}-\left(x^{2} y-x-1\right)^{2}\) has only two critical points, but has local maxima at both of them. Then produce a graph with a carefully chosen domain and viewpoint to see how this is possible.

If a function of one variable is continuous on an interval and has only one critical number, then a local maximum has to be an absolute maximum. But this is not true for functions of two variables. Show that the function \(f(x, y)=3 x e^{y}-x^{3}-e^{3 y}\) has exactly one critical point and that f has a local maximum there that is not an absolute maximum. Produce a graph with a carefully chosen domain and viewpoint to see how this is possible. Equation Transcription: Text Transcription: f(x,y)=3xe^y-x^3-e^3y

Find the shortest distance from the point \((2,0,-3)\) to the plane \(x+y+z=1\). Equation Transcription: Text Transcription: (2,0, -3) x+y+z=1

Find the point on the plane \(x-2 y+3 z=6\) that is closest to the point \((0,1,1)\). Equation Transcription: Text Transcription:

Find the points on the cone \(z^{2}=x^{2}+y^{2}\) that are closest to the point \((4,2,0)\). Equation Transcription: Text Transcription: z^2=x^2+y^2 (4,2,0)

Find the points on the surface \(y^{2}=9+x z\) that are closest to the origin. Equation Transcription: Text Transcription: y^2=9+xz

. Find three positive numbers whose sum is 100 and whose product is a maximum.

Find three positive numbers whose sum is \(12\) and the sum of whose squares is as small as possible. Equation Transcription: Text Transcription: 12

Find the maximum volume of a rectangular box that is inscribed in a sphere of radius \(r\). Equation Transcription: Text Transcription: r

Find the dimensions of the box with volume \(1000 \mathrm{~cm}^{3}\) that has minimal surface area. Equation Transcription: Text Transcription: 1000 cm^3

Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane \(x+2 y+3 z=6\). Equation Transcription: Text Transcription: x+2y+3z=6

Find the dimensions of the rectangular box with largest volume if the total surface area is given as \(64 \mathrm{~cm}^{2}\). Equation Transcription: Text Transcription: 64 cm^2

Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its \(12\) edges is a constant \(c\). Equation Transcription: Text Transcription: 12 c

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. Equation Transcription: Text Transcription: V

A cardboard box without a lid is to have a volume of \(32,000 \mathrm{~cm}^{3}\). Find the dimensions that minimize the amount of cardboard used. Equation Transcription: Text Transcription: 32,000 cm^3

A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a rate of \(10 \text { units/m }^{2}\) per day, the north and south walls at a rate of \(8 \text { units/m }^{2}\) per day, the floor at a rate of \(1 \text { units/m }^{2}\) per day, and the roof at a rate of \(5 \text { units/m }^{2}\) per day. Each wall must be at least \(30 m\) long, the height must be at least \(4 m\), and the volume must be exactly \(4000 \mathrm{~m}^{3}\). (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 boundary of the domain.) (c) Could you design a building with even less heat loss if the restrictions on the lengths of the walls were removed? Equation Transcription: Text Transcription: 10 units/m^2 8 units/m^2 1 units/m^2 5 units/m^2 30 m 4 m 4000 m^3

If the length of the diagonal of a rectangular box must be \(L\), what is the largest possible volume? Equation Transcription: Text Transcription: L

A model for the yield \(Y\) of an agricultural crop as a function of the nitrogen level \(N\) and phosphorus level \(P\) in the soil (measured in appropriate units) is \(Y\left(N_{r} P\right)=k N P e^{-N-P}\) where \(k\) is a positive constant. What levels of nitrogen and phosphorus result in the best yield? Equation Transcription: Text Transcription: Y N P Y(N,P)=kNPe^-N-P k

The Shannon index (sometimes called the Shannon-Wiener index or Shannon-Weaver index) is a measure of diversity in an ecosystem. For the case of three species, it is defined as \(H=-p_{1} \ln p_{1}-p_{2} \ln p_{2}-p_{3} \ln p_{3}\) where \(p_{i}\) is the proportion of species \(i\) in the ecosystem. (a) Express \(H\) as a function of two variables using the fact that \(p_{1}+p_{2}+p_{3}=1\). (b) What is the domain of \(H\) ? (c) Find the maximum value of \(H\). For what values of \(p_{1}, p_{2}, p_{3}\) does it occur? Equation Transcription: Text Transcription: H=-p_1 ln p_1-p_2 ln p_2-p_3 ln p_3 p_1 i H p_1+p_2+p_3=1 p_1,p_2,p_3

Three alleles (alternative versions of a gene) \(A, B\), and \(O\) determine the four blood types \(A (AA or AO)\), \(B (BB or BO)\), \(O (OO)\), and \(AB\). The Hardy-Weinberg Law states that the proportion of individuals in a population who carry two different alleles is \(P=2 p q+2 p r+2 r q\) where \(p, q\), and \(r\) are the proportions of \(A, B\), and \(O\) in the population. Use the fact that \(p+q+r=1\) to show that \(P\) is at most \(\frac{2}{3}\). Equation Transcription: Text Transcription: A,B,O A(AA or AO) B(BB or BO) O(OO) AB P=2pq+2pr+2rq p,q, and r p+q+r=1 P 2/3

Method of Least Squares Suppose that a scientist has reason to believe that two quantities \(x\) and \(y\) are related linearly, that is, \(y=m x+b\), at least approximately, for some values of \(m\) and \(b\). The scientist performs an experiment and collects data in the form of points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\), and then plots these points. The points don’t lie exactly on a straight line, so the scientist wants to find constants \(m\) and \(b\) so that the line \(y=m x+b\) “fits” the points as well as possible (see the figure). Let \(d_{i}=y_{i}-\left(m x_{i}+b\right)\) be the vertical deviation of the point \(\left(x_{i}, y_{i}\right)\) from the line. The method of least squares determines \(m\) and \(b\) so as to minimize \(\sum_{i=1}^{n} d_{i}^{2}\), the sum of the squares of these deviations. Show that, according to this method, the line of best fit is obtained when \(m \sum_{i=1}^{n} x_{i}+b n=\sum_{i=1}^{n} y_{i}\) and \(m \sum_{i=1}^{n} x_{i}^{2}+b \sum_{i=1}^{n} x_{i}=\sum_{i=1}^{n} x_{i} y_{i}\) Thus the line is found by solving these two equations in the two unknowns \(m\) and \(b\). (See Section \(1.2\) for a further discussion and applications of the method of least squares.) Equation Transcription: Text Transcription: x y y=mx+b m b (x_1,y_1),(x_2,y_2), ldots,(x_n,y_n) d_i = y_i - (mx_i + b) (x_i,y_i) sum_i=1 ^n d_i ^2 m sum_i=1 ^n x_i + bn = sum_i=1 ^n y_i m sum i=1 ^n x_i ^2 + b sum i=1 ^n x_i = sum_i=1 ^n x_i y_i 1.2

Find an equation of the plane that passes through the point \((1,2,3)\) and cuts off the smallest volume in the first octant. Equation Transcription: Text Transcription: (1,2,3)