Suppose f ( x , y )= x ^2+ y^ 22 x 4 y +3 (A) How many critical points does f ha

ID: 2851390 • Letter: S

Question

Suppose f(x,y)=x^2+y^22x4y+3

(A) How many critical points does f have in R2?

(B) If there is a local minimum, what is the value of the discriminant D at that point? If there is none, type N.

(D) If there is a saddle point, what is the value of the discriminant D at that point? If there is none, type N.

(E) What is the maximum value of f on R2? If there is none, type N.

(F) What is the minimum value of f on R2? If there is none, type N.

f(x,y)=x2+y22x4y+3

==> fx (x,y) = 2x+ (0) 2(1)(0)+(0) = 2x -2

==> fy (x,y) = 0+ 2y 04(1) +(0) = 2y -4

critical points ==> fx = 0 , fy = 0

==> 2x -2 = 0 ==> x = 1 , 2y -4 = 0 ==> y = 2

Hence critical points = (1,2)

fxx = 2 , fyy = 2 , fxy = 0

D = fxxfyy - (fxy)2

==> D = 2(2) - (0)2 = 4 > 0

and fxx > 0

Hence local minimum at (1,2)

No local minima or saddle points

f(1,2) = (1)2+(2)22(1)4(2)+3 = -2 minimum value

No maximim value