9/ x 2 - 2x + 1 7 3. (12 points) Suppose that the temperature at the point (x, y) in plane R2 is given by T(x, y) = 4(x - 1)2 + 2y2, measured in degrees Celsius. (a) Using the method of Lagrange Multipliers, find the maximum temperature T(z, y) on the circle x2 + y2 = 1. Show all work for partial credit. V ( ( x y, z ) = 2 vg (x y, z ) g(xy ) = x ty? = 1 T ( x , y ) = 4 x 2 - 8 * + 4 + 2g2 0 9( x,y ) : 8 x- 8 = 2 2x- 2%= 4x - 4 4 y = 2 24 7 2 =2 -7 4 x-4=2x *2 +42=1 2 x = 4 X = 2 cost, sint 4 = 1 - 4 = - 3 8 x - 8 = 2sin x 8x-8 = - 2.0- 4 y = 2 2 casy 2= 24 dead ovodo Jolq efT (1) cosy cosxtsing = 1 bled be (b) On the graph of x2 + y2 = 1 below, indicate the point(s) at which the temperature T is maximum and sketch the level curve of T(x, y) which corresponds to the maximum value of T(x, y). A rough an sketch is fine, as long as proportions and orientation are reasonable. 73 e i d evoda gold Lter to any 4