Question

4. Find the maximum and minimum values of f(x, y) = 4x2 + 10y2 on the disk x2 + y2 < 4.

Answer 1

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Hing first find all the critical points that are in the disk Here are the two first order partial derivatives fx = 8 x 7 8x=0 = 2=0 fy = goy g oyco - Y=0 So the only critical point is (0,0) and it does satisfy the inequality. At the point we proceed with Lagrange Multipliers and We treat the Constraint so, Here is the system of equations that we need to Solve 8= 2da goy - ady x2 +42 - 4 From the first equation we get 2x (4 od) =0 x=0 ore d = 4 It we have a=0 then the constraint gives us y = 12 If we have de 4 the second equation gives us goy = 84 - Y=0 The Constraint then tells us that a = 22 If we'd performed a similar analysis on the second equation We would arrive at the same points. so Lagrange Multipliers gives us four point to Check (0,2), (0-2), (2,0), (-2,0) To find the manimum and minimum we need to simply plug these four points along with the Caitical point in the function f(0,0) = 0 minimum f (9,0) = f(-2,0) = 16 f(0, 2) = f(0, 2) = 40 manimum In this case, the minimum was interior to disk and the manimum was on the boundary of the disk.