Math 16B, Section 1Fall 2005SarasonREVIEW EXERCISES 11. In each part, find the first partial derivatives of the given functionf(x, y)(a)f(x, y) =x5-10x3y2+ 5xy4(b)f(x, y) = 20x3/5y2/5(c)f(x, y) =e-x2/y2. In each part, evaluate the integralI=Rf(x, y)dydxof the given functionfover the givenregionR.(a)f(x, y) =x2+y4;Ris the rectangle given by-1≤x≤1, 0≤y≤2.(b)f(x, y) =x2-y2;Ris the triangle with vertices (0,0),(1,0),(1,2).(c)f(x, y) =x2y3;Ris defined by the inequalitiesx2+y2≤1,y≥0.3. For each part, determine the critical points of the functionf(x, y), and determine the natureof each critical point, to the extent possible, by means of the second derivative test.(a)f(x, y) =x3-y2-3x+ 4y(b)f(x, y) =x3+y3-9xy(c)f(x, y) = 4x4-2y2-xy(d)f(x, y) = 4x8-8x4-xy4. (a) Find the maximum and minimum values of the functionf(x, y) = 7x+ 9yon the curve7x4+ 9y4= 64.(b) Where does the functionf(x, y) =x2+ 3y2+ 10 attain its minimum value under theconstraintx+y= 8?
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Sarason REVIEW EXERCISES, function x2, Florida Anti-Intellectual League, Central American subsidiary