Quiz 2 for MATH 105 SECTION 205
January 21, 2015
Given Name
Family Name
Student Number
1. (a) (1 point) Let f (x, y) = y sin(xy), compute fyx (x, y).
(a)
(b) (1 point) Can you find a function h(x, y) such that hxx = y 2 cos(xy 2 ) and hxy = xy 2 cos(xy 2 )? (Do not
need to find h(x, y), just put ‘Yes’ or ‘No’)
(b)
2. (a) (0.5 points) Let g(x, y) = 2x2 + y 4 + 1, find a local minimum point of g(x, y).
(a)
(b) (1 point) Let R = {(x, y) :
f (x, y) on R.
x2
+
2y 2
≤ 9} and f (x, y) =
x2
+ 4y 2 , find the absolute maximum value of
(b)
3. Let R = {(x, y) : x2 + y 2 ≤ 4} and f (x, y) = 2x2 + 2y 2 + 1, then
(a) (1 point) Find the absolute maximum value of f (x, y) on R.
(a)
(b) (0.5 points) Find the absolute minimum value of f (x, y) on R.
(b)
4. Let f (x, y) = x2 + xy 2 − 2x + 1, then
(a) (2 points) Find all critical points of f (x, y).
(a)
(b) (1 point) Compute the Hessian matrix of f (x, y).
(b)
(c) (1 point) Compute the discriminant D(x, y) of f (x, y).
(c)
(d) (2 points) Classify all critical points of f (x, y). (Make a table)
Your Score:
/11