Student Name:
Exam 2 - Math 4320/5320 - Modern Analysis II
Dr. Radu C. Cascaval, Spring 2016
Take Home - Due Thursday, April 28 before class time
You may consult the textbook or other printed materials (use of internet resources is highly discouraged),
but you cannot get help from any breathing source (like humans, monkeys, dogs etc).!
1. Consider the function

2
2
 xy x − y , (x, y) 6= (0, 0)
f (x) =
x2 + y 2

0, (x, y) = (0, 0)
(a) Show that f is continuously on R2 .
(b) Show that f is differentiable on R2 .
(c) Show that fxy (0, 0) and fyx (0, 0) exist but are not equal. What does this say about the
continuity of fxy and fyx ?
2. Let f : R2 → R be defined by
f (x) =

 (x2 + y 2 ) sin

x2
1
, (x, y) 6= (0, 0)
+ y2
0, (x, y) = (0, 0)
(a) Show that f is continuous at (0, 0), both partial derivatives exist but none of the partial
derivatives of f is bounded in a neighborhood of (0, 0).
(b) Show that f is differentiable, but none of the partial derivatives of f is continuous at (0, 0).
3. Find all the local extrema of the functions
f (x, y) = x3 y 2 (6 − x − y) and g(x, y) = ex
2 −4y+y 2
Justify your findings!
4. Using the Lagrange Multiplier Method, find the point on the intersection of the two planes given
by x + y + z = 1 and 3x + 2y + z = 6 that is closest to the origin in R3 .
5. [Math 5320 only]. Let F : R2 → R2 be defined by F (x, y) = (x2 − y 2 , −2xy + 1). Prove that there
exists an > 0 such that, for all fixed (u, v) with |(u + 1, v − 1)| < , there exists exactly two
distinct points (x1 , y1 ) and (x2 , y2 ) so that F (xj , yj ) = (u, v) for j = 1, 2. Why and how does the
Inverse Function Theorem fail in this case?