ID: Find all the local maxima, local minima, and saddle points of the function.
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1 1. Find all the local maxima, local minima, and saddle points of the function. f(x,y) = x + xy + y + 5x 5y + 4 A. A local maximum occurs at. The local maximum value(s) is/are. B. There are no local maxima. A. A local minimum occurs at. The local minimum value(s) is/are. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. A saddle point occurs at. Answers B. There are no local maxima. A. A local minimum occurs at ( 5,5). The local minimum value(s) is/are 21. ID:
2 2. Find all the local maxima, local minima, and saddle points of the function shown below. f(x,y) = 2 5 x 2 + y 2 Find the local maxima. Select the correct choice below and, if necessary, fill in the answer boxes as needed to complete your choice. A. A local maximum occurs at. The local maximum value(s) is/are. B. There are no local maxima. Find the local minima. Select the correct choice below and, if necessary, fill in the answer boxes as needed to complete your choice. A. A local minimum occurs at. The local minimum value(s) is/are. Find the saddle points. Select the correct choice below and, if necessary, fill in the answer box as needed to complete your choice. A. A saddle point occurs at. Answers A. A local maximum occurs at (0,0). The local maximum value(s) is/are 2. ID:
3 3. Find all the local maxima, local minima, and saddle points of the function. y f(x,y) = e (x + y ) + 1 A. A local maximum occurs at. The local maximum value(s) is/are. B. There are no local maxima. A. A local minimum occurs at. The local minimum value(s) is/are. (Type anexact answer. Use a comma to separate answers as needed.) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. A saddle point occurs at. Answers B. There are no local maxima. A. A local minimum occurs at (0,0). The local minimum value(s) is/are 1. (Type anexact answer. Use a comma to separate answers as needed.) A. A saddle point occurs at (0,2). ID: Find the absolute maximum and minimum of the function on the given domain. f(x,y) = 4x + 5y on the closed triangular plate bounded by the lines x = 0, y = 0, y + 2x = 2 in the first quadrant The absolute maximum is. The absolute minimum is. Answers 20 0 ID:
4 5. Find the absolute maximum and minimum of the function π π rectangular plate 1 x 3, y f(x,y) = (48x 12x ) cos y on the 2 z=(48x 12x ) cos y The absolute maximum is. The absolute minimum is. Answers ID: Find the point on the plane 4 x + 3 y + z = 10 that is nearest the origin. What are the values of x, y, and z for the point? x = y = z = (Type integers or simplified fractions.) Answers ID: Find three numbers whose sum is 30 and whose sum of squares is a minimum. The three numbers are. (Use a comma to separate answers as needed.) Answer: 10,10,10 ID:
5 8. Find the extreme values of f(x,y) = xy subject to the constraint x + y 24 = 0. The extrema are. (Use a comma to separate answers as needed. Simplify your answer.) Answer: 12, 12 ID: Use the method of Lagrange multipliers to find a. the minimum value of x + y, subject to the constraints xy = 4, x > 0, y > 0. b. the maximum value of xy, subject to the constraint x + y, = 4. The minimum value of x + y is. The maximum value of xy is. Answers 4 4 ID: The temperature at point (x,y) on a metal plate is T(x,y) = 256x 256xy + 64y. An ant on the plate walks around the circle of radius 5 centered at the origin. What are the highest and lowest temperatures encountered by the ant? The minimum temperature is The maximum temperature is degrees. degrees. Answers ID:
6 11. Find the point on the sphere x 2 + y 2 + z 2 = 16 farthest from the point ( 1, 1, 1). The point is (,, ). (Type an ordered triple.) Answers ID: Find three real numbers x, y, and z whose sum is 12 and the sum of whose squares is as small as possible. The three numbers are. (Simplify your answers. Use a comma to separate answers as needed.) Answer: 4,4,4 ID: Find the dimensions of the closed rectangular box with maximum volume that can be inscribed in the unit sphere. The dimensions are. (Type exact answers, using radicals as needed. Use a comma to separate answers as needed.) Answer: ,, ID:
7 14. 2 A space probe in the shape of the ellipsoid 36x + y + 4z = 41 enters a planet's atmosphere and its surface begins to 2 heat. After 1 hour, the temperature at the point (x,y,z) on the probe's surface is T(x,y,z) = 72x + 4yz 16z Find the hottest point on the probe's surface. The hottest point is ( ±,, ). (Simplify your answer. Type exact answers, using radicals as needed. Use integers or fractions for any numbers in the expression.) Answers ID:
Find the specific function values. Complete parts (a) through (d) below. f (x,y,z) = x y y 2 + z = (Simplify your answer.) ID: 14.1.
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