4. Write the second degree Taylor polynomial for the function f(x, y, z) = xy + yz + xz at (1, 2, 3).

5. Categorize the quadratic form p(x, y, z) = x 2 + 5y 2 + 3z 2 + 4xy + 3xz + yz as positive definite, negative definite or indefinite.

6. Use the method of Lagrange multipliers to find the minimum value of f(x, y, z) = x 2 + y 2 + z 2 subject to z = x y 6.

7. Use the chain rule to find the derivative of gf at a = (3, 2) if f(x, y) = (y, x) and g(x, y) = (x 2 y 3 , 3x y 2 ).

8. Evaluate lim (x,y)(0,0) 1 e 2x3y sin(6y 4x) or show that the limit does not exist.