Second Derivative Test for Functions of Two Variables

Question

1. Let f(x,y)=x²+y²+8x-12y+13
2. Find f_x(x,y)
3. find f_y(x,y)
4. find f_xx(x,y)
5. find f_yy(x,y)
6. find f_xy(x,y)
7. Use parts (a) and (b) to find the critical point of f.
8. Determine whether the critical point is a local maximum, a local minimum, or a saddle point. Use the Second Derivative test for Functions of Two Variables to find your answer. Show your work

Answer 1

1. Let f(x,y)=x²+y²+8x-12y+13
2. Find f_x(x,y)
3. find f_y(x,y)
4. find f_xx(x,y)
5. find f_yy(x,y)
6. find f_xy(x,y)
7. Use parts (a) and (b) to find the critical point of f.
8. Determine whether the critical point is a local maximum, a local minimum, or a saddle point. Use the Second Derivative test for Functions of Two Variables to find your answer. Show your work

a) f_x(x,y)=(x²+y²+8x-12y+13)_x=2x+8

b) f_y(x,y)=(x²+y²+8x-12y+13)_y=2y-12

c) f_xx(x,y)=(2x+8)_x=2

d) f_yy(x,y)=(2y-12)_y=2

e) f_xy(x,y)=(2y-12)_x=0

f) f_x(x,y)=f_y(x,y)=0=2x+8=2y-12

critical point (-4,6)

g) D(-4,6)=f_xx(-4,6)f_yy(-4,6)-[f_xy(-4,6)]²=(2)(2)-(0)²=4

Since D>0 and f_xx(-4,6)>0, f(-4,6) is a local minimum