The temperature at a point (x,y,z) of a solid E bounded by the coordinate planes and the plane x+y+z=1 is T(x, y, z) = (-9xy + 8z +20) degrees Celsius. Find the average temperature over the solid. (Give the exact answer.)
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- FInd the absolute extrema (maximum and minimum of the function on the region bounded by the square with vertices at (0,0),(7,0),(0,7), and (7,7)Find the local minimum, maximum, and saddle points – if any – of the function f(x, y) = 2x^2 + y^2 + 2x^2 y. Find the maximum and minimum values of the function f(x, y) = xy on the closed and bounded region defined as {(x, y) | x^2 + y^2 ≤ 1}. [Hint: You need to analyze the function first on the open region {(x, y) | x^2 + y^2 < 1} and then on the boundary {(x, y) | x^2 + y^2 = 1}.]Consider the function: f(x, y) = 2x3 + xy2 + 5x2 + y2 + 5 Find all the critical point of f. Use the 2nd derivative test to classify the critical points of f. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f has local maximum value(s) at (x,y) = f has local minimum value(s) at (x,y) = f has saddle point(s) at (x,y) =
- Identify any extrema of the function by recognizing its given form or its form after completing the square. Verify your results by using the partial derivatives to locate any critical points and test for relative extrema. (If an answer does not exist, enter DNE.) f(x, y) = square root of x2 + y2 + 64 relative minimum (x, y, z) = relative maximum (x, y, z) =Find the critical points for the function f(x,y)=x3+y3−3x2−3y+2 and classify each as a local maximum, local minimum, saddle point, or none of these. critical points: (give your points as a comma separated list of (x,y) coordinates.) classifications: (give your answers in a comma separated list, specifying maximum, minimum, saddle point, or none for each, in the same order as you entered your critical points)A differentiable function f has one critical number at x = 2. Identify the relative extrema of f at the critical number when f′(1) = 2 and f′(3) = 6.
- Identify any extrema of the function by recognizing its given form or its form after completing the square. Verify your results by using the partial derivatives to locate any critical points and test for relative extrema. g(x,y) = (x-6)^2 + (y-2)^2 relative minimum (x,y,z) = relative maximum (x,y,z) =Examine the function for relative extrema. f(x, y) = |x + y| − 5 (x, y, z) =A raptured pipe in a North Sea oil rig produces a circlar oil slick that is y meters thick at a distance x meters from the rapture. Turbulance makes it difficult to directly measure the the thickness of the slick at the source (where x=0), but for x >0, it is found that y= 0.5(x2+3x)/x3+x2+4x. Assuming the oil slick is continiously distributed, how thick would you expect it to be at the source?