Question

5

point and us approximate f(3.05, 3.95). 5. Find the local minimum and maximum values and saddle points of f(x, y) = -4x2 - y2 + 16x – 2y + 1 if any. 6. Find panations of th

Answer 1

= Solution: 5)Given function : f(x,y)=-4x2 – 2,2 +16x-2y+1 Critical points are where to = 0 and 1, = 0 D = (-4x2 - y2 +16x – 2y+1)= -8x+16 a (-4x2 - y2 +16x-2y+1)=-2y-2 fx = 0 =-8x+16= 0 = x= 2 →(1) 1, = 0 =-2y-2=0=y=-1 —(2) =>x=2, y =-l is the only critical point =(2,-1) is the critial point = Only one critical point for f (x,y) To characterize each critical point: to = -8x+16 = x= -8 Jy = -2y-2= $y = -2 and For (x,y)=(2,-1) = m =-8, and So Hessian D= for yn - (Tv) = D2-) = 16– 0 = 16 >0 Since D(a-n = 16 >0 and since fax and <0,(2,-1) is a Maximum point Since there is only one critical point which is a local minimum point, there is NO local maximum for the given function. Since there is only one critical point which is a local minimum point, there is NO Saddle Point for the given function. Maximum value = f (2,-1)=-4(2)*-(-11° +16(2)-2(-1)+1=18 =(2, -1) = 18 is the local maximum value of the given function F = 0 S = 0 tym = -2 ענן'