The sphere 2+ y+z = 36 and the plane x + y + z = -10...

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The sphere 2+ y+z = 36 and the plane x + y + z = -10 intersect in an ellipse. Use Lagrange multipliers, or any other method, to find the lowest point (= point with minimal z-value) on this ellipse. QUESTION 2 Find the absolute maximum and absolute minimum values of the function f(x, y) = (x-1)+y-3y on the region {(x, y) = R : (x 1) + (y-1) 9}. Hint: to find the maximum and minimum on the boundary of this region, you might want to use Lagrange multipliers. QUESTION 3 Find the absolute maximum and absolute minimum values of the function f(x, y) = x + y + z on the region {(x, y, z) = R : x + y + z <1}. Hint: to find the maximum and minimum on the boundary of this region, you might want to use Lagrange multipliers. QUESTION 4 Find all complex solutions to the equation w 64 (that is, find all of the third roots of -64.) Express your answers in the form a + bi. QUESTION 5 Find all solutions in complex numbers to the equation w-4w+5=0. Express your answers in the form a + bi. Find the complex number QUESTION 6 Express your answer in the form a + bi. QUESTION 7 Find all solutions in complex numbers to the equation w+6w+5=0. Express your answers in the form a + bi. The sphere 2+ y+z = 36 and the plane x + y + z = -10 intersect in an ellipse. Use Lagrange multipliers, or any other method, to find the lowest point (= point with minimal z-value) on this ellipse. QUESTION 2 Find the absolute maximum and absolute minimum values of the function f(x, y) = (x-1)+y-3y on the region {(x, y) = R : (x 1) + (y-1) 9}. Hint: to find the maximum and minimum on the boundary of this region, you might want to use Lagrange multipliers. QUESTION 3 Find the absolute maximum and absolute minimum values of the function f(x, y) = x + y + z on the region {(x, y, z) = R : x + y + z <1}. Hint: to find the maximum and minimum on the boundary of this region, you might want to use Lagrange multipliers. QUESTION 4 Find all complex solutions to the equation w 64 (that is, find all of the third roots of -64.) Express your answers in the form a + bi. QUESTION 5 Find all solutions in complex numbers to the equation w-4w+5=0. Express your answers in the form a + bi. Find the complex number QUESTION 6 Express your answer in the form a + bi. QUESTION 7 Find all solutions in complex numbers to the equation w+6w+5=0. Express your answers in the form a + bi.