Homework 2
Exercise 1. Compute direction derivative of the following functions at the point (0, 0, 1) in the
direction (0, 2, 1)
1) f (x, y, z) = −2x2 − 2xy − xz − 21 y 2 + 2xz − 2z 2 + x − 2y − z
2) f (x, y, z) = (x + 2y + 3z)2
3) f (x, y, z) = x2 + x2 y + y 2 z + z 2 − 4z
Exercise 2. Determine the definiteness of the following matrices:
2 −7
(a)
3 −8
2 4
(b)
−2 6


2 1 −1
(c) 0 1 1 
2 0 −2


1 −1 0
(d) −1 2 −1
0 −1 1
Exercise 3. a) What is the matrix associated to the quadratic form
Q(x, y) = 3x2 − (5 + c)xy + 2cy 2
b) What is the value of c for that the associated Hessian matrix is
(i) positive definite, (ii) positive semidefinite
Exercise 4. Determine which of the following sets are convex by drawing each in the plane.
(a) {(x, y) | x2 + y 2 < 2};
(b) {(x, y) | x ≥ 0, y ≥ 0};
(c) {(x, y) | x2 + y 2 > 8};
(d) {(x, y) | x ≥ 0, y ≥ 0, xy ≥ 1};
(e) {(x, y) | xy ≤ 1};
√
√
(f ) {(x, y) | x + y ≤ 2};
Exercise 5. Give the boundary of the following sets. Writing the answer is enough. However,
for two last points, I need you to explain a bit.
1. Rn
2. Rn+
3. Rn++
4. B̄(0, r) for r > 0
1
5. B(0, r) for r > 0
6. in R2 , [1, 2] × {0}
7. in R, [1, 2]
8. in R, { k1 | k = 1, 2, ...} = {1, 12 , ..., n1 , ..., }
9. a linear subspace of Rn but not Rn (A bit difficult!)
Exercise 6. (a) Let E be the subset in R2 consisting of the point (0, 0) and all point of the
form (1/n, 1/m) for n = 1, 2, . . . and m = 1, 2, . . . Is E closed?
(b) Let F be the subset in R2 defined by F = {(0, 0)} ∪ {(1/n, 1/n) | n = 1, 2, . . .}. Is F closed?
Exercise 7. Consider the following three subsets of R2 :
A = {(x, y) : y = 1, x ∈ ∪∞
n=1 (2n, 2n + 1)}
B = {(x, y) : y ∈ (0, 1), x ∈ ∪∞
n=1 (2n, 2n + 1)}
C = {(x, y) : y = 1, x ∈ ∪∞
n=1 [2n, 2n + 1]}
For each of these sets determine whether it is open, closed, or neither.
2