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final-practice2

final-practice2 - Problem 1 Find the critical points of the...

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Problem 1. Find the critical points of the function f ( x, y ) = 2 x 3 - 3 x 2 y - 12 x 2 - 3 y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Problem 2. Determine the global max and min of the function f ( x, y ) = x 2 - 2 x + 2 y 2 - 2 y + 2 xy over the compact region - 1 x 1 , 0 y 2 . Problem 3. Using Lagrange multipliers, optimize the function f ( x, y ) = x 2 + ( y + 1) 2 subject to the constraint 2 x 2 + ( y - 1) 2 18 . Problem 4. Consider the function w = e x 2 y where x = u v, y = 1 uv 2 . Using the chain rule, compute the derivatives ∂w ∂u , ∂w ∂v . Problem 5. (i) For what value of the parameter a , will the planes ax + 3 y - 4 z = 2 , x - ay + 2 z = 5 be perpendicular? (ii) Find a vector parallel to the line of intersection of the planes x - y + 2 z = 2 , 3 x - y + 2 z = 1 . (iii) Find the plane through the origin parallel to z = 4 x - 3 y + 8 . (iv) Find the angle between the vectors v = (1 , - 1 , 2) , w = (1 , 3 , 0) . 1
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(v) A plane has equation z = 5 x - 2 y + 7 .
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