EXAM 02 –gilbert –(55035)1This print-out should have 14 questions. Multiple-choice questions may continue on the next column or page –find all choices before answering.001 10.0 pointsDetermine the maximum value off (x, y) = 8x2/3y1/3subject to the constraintg(x, y) = 2x +y −6 = 0.1. maximum = 14 2. maximum = 17 3. maximum = 15 4. maximum = 18 5. maximum = 16 correct Explanation:By the Method of Lagrange multipliers, the extreme values occur at the common solutions of(∇f)(x, y) = λ(∇g)(x, y) , g(x, y) = 0 .Now∂f∂x=163x−1/3y1/3,∂f∂y=163x−1/3y1/3.Thus(∇f)(x, y) = D163x−1/3y1/3,38x2/3y−2/3E .On the other hand,(∇g)(x, y) = h2,1 O.But then by the condition on ∇fand ∇g,163x−1/3y1/3= 2λ ,83x2/3y−2/3=λ ,which after simplification givesλ =83x−1/3y1/3=83x2/3y−2/3,I.E., y=x. Thus by the constraint equation,g (x, x) = 2x +x −6 = 0,I.E., x= 2. Consequently,( 2,2 ) ,is a critical point at whichf (2, 2) = 8(2)2/3(2)1/3= 16.And so fhasmaximum value = 16subject to the constraint g(x, y) = 0.keywords:002 10.0 pointsThe graph of g(x, y) = 0 is shown as a dashed line in123321-0y-1-2-3xP-3-2-1while the level curves f(x, y) = kfor a func-tion z= f(x, y) are shown as continuous curves with values of klisted at the edges.Which one of the following properties does fhave at Psubject to the constraint g(x, y) = 0?1. a local max at P
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