f(x1, x2) = x4 + x4
1 2
2 2
1 2
f(x1, x2) = ex + ex - x200 - x200
1 2
f(x1, x2) = 2x2 - 8x1x2 + x2
1 2
f(x1, x2) = 4x2 + 2x1x2 + 2x2
1 2
f(x1, x2, x3) = x3 + x3 + x2
1 2 3
f(x1, x2) = x2 - 2x1x2 + x4
1 2 2
xT Ax
f(x) =
||x|| + 1


8 -2
f(x) = xT x - 3 -2, -3 x
0 7

f(x1, x2) = (x4 - x2)2
1
f(x1, x2, x3) = x4 - 2x2 + x2 + 2x2x3 + 2x2
1 1 2 3
f(x1, x2, x3) = 4x2 + 4x1x2 + 4x2 + 2x1x3 + 4x2x3
1 2
f(x1, x2) = 2x3 - 6x2 + 3x2x2
2 2 1
f(x1, x2) = x4 + 2x2x2 + x2 - 4x2 - 8x1 - 8x2
1 1 2 1
f(x1, x2) = (x1 - 2x2)4 + 64x1x2
f(x1, x2) = 2x2 + 3x2 - 2x1x2 + 2x1 - 3x2
1 2
f(x1, x2) = x2 + 4x1x2 + 2x2 + x1x2
1 2
x1 + x2
f(x) =
2
x2 + + 1
1 2
x

-3 2
1
f(x) = xT x + -2, -1 x + e2
2
0 -2


1 0
f(x) = xT x + 4, 6 x + e-2
3 -5
d ² > 0
²d
f : R2 R
f(x) min
x1, x2 e" 0
"f "f
x = [x1, x2]T "f(0) = 0 (0) d" 0 (0) d" 0 0

"x1 "x2

d ² > 0
²d



A = {(x1, x2) : 3x1 + x2 d" 3, 3x1 - x2 e" -2, x1 e" 0}
B = {(x1, x2) : x1x2 d" 2, x1 e" 0}
C = {(x1, x2) : x1 - x2 d" 2, x2 + x2 d" 9}
1 2
D = {(x1, x2) : 0 < |x1| + |x2 - 2| d" 9}
E = {(x1, x2, x3) : x3 e" x2 + x2}
1 2
F = {(x1, x2, x3) : x2 d" x2 + x2}
3 1 2
2
G = {(x1, x2, x3) : + x2 d" 16}
nx1 2
H = {x " Rn : x2 d" 1}
n i=1 i
I = {x " Rn : x2 e" 1}
i=1 i
J = {x " Rn : maxxi d" 1}
i
K = {x " Rn : minxi e" 1}
i
"
2
1
L = {(x1, x2) : ex -x2 d"
"7}
2
= {(x1, x2) : (x2 + x2) ex d" 16, x2 + 4 e" x2}
1 1
x1
2
M = {(x1, x2) : (x1 + x2)e d" 4, x2 d" x1 + 4, x1 > 0}
2 2
x1
N = {(x1, x2) : 3 ln + 2 ln x2 + ln(12 - x1 - x2) e" 2, x1 > 0, x2 > 0, x1 + x2 < 12}
6

f(x1, x2) = x1x2 x1, x2 > 0
f(x1, x2, x3) = 5x2 + 5x2 + 4x2 + 2x2x3 + 4x1x2
1 2 3
f(x1, x2, x3) = x2 + x1x2 + 5x2 + x1x3 + 4x2x3
1 2
1
f(x1, x2) = x1, x2 > 0
x1x2
x1
f(x1, x2) = x1, x2 > 0
x2
x2
1
f(x1, x2) = x1, x2 > 0
x2
f(x1, x2) = xÄ…xÄ…-1 x1, x2 > 0 Ä… d" 1
1 2
2
1
f(x) = ex +4x2 x " Rn
f(x) = ln(x2 x2) x2 - x2 > 0
1 1
-

6 4
1
f(x) = xT x + 5, 7 x
2
2 -2

"
2 2
f(x) = xT x + 5, ln 6 x
3 4
f : Rn R {x " Rn : f(x) e" c}
c
f : Rn R {x " Rn : f(x) = c}
c
{fn} M

f(x) = lim fn(x)
n"
x " M f


I = [a, b] ‚" R M
Rn f : (t, x) f(t, x) I × M
I x " M M
t " I
b
F (x) = f(t, x)dt
a
M
f : M R g : R R M ‚" Rn X ‚" R
f(M) ‚" X f g
g ć% f f g g ć% f






6 8
4x1 + x2 max xT x min
2
0 6
x2 + x2 = 9
1 2
||x||2 = 2

3 4
1
x1x2 max xT x
2
0 3
x2 + 4x2 = 1
1 2
||x||2 = 1
x3 + 3x2 + x3 x1x2x3
1 2

x2 + x2 + x2 = 16 x2 + x2 + x2 = 12
1 2 3 1 2 3
x1
x2 + x2 max
1 2
x2 + 2x2 + 2x2 = 8
1 2 3
3x2 + 4x1x2 + 6x2 = 140
1 2
x3 = x1 + x2
2x1 + 3x2 - 4 min x1 + x2 + 2x3

x1x2 = 6 x2 + x2 + x2 = 3
1 2 3

9
(x1 - )2 + (x2 - 2)2 min
4 1
e-(x +x2) min
x2 - x2 d" 0
1
1 2
e-x + e-x d" 10
x1 + x2 - 6 d" 0
x2 + x2 - 4x1 - 4x2 min (x2 - x1)2 + (1 - x1)2 + 1 min
1 2
x2 d" x2 x2 - x1 - 1 d" 0
1
x1 + x2 d" 2 -x2 - x1 + 1 d" 0
(x2 - x1)2 + (1 - x1)2 + 1 min
x3 + 3x2 + x3 min
1 2
x2 - x1 - 1 d" 0
x2 + x2 + x2 = 16
1 2 3
-x2 - x1 + 3 d" 0
x2 + x2 min x3 - 6x1 + x3 - 3x2 min
1 2 1 2
x2 + 1 d" x1 x1 + x2 d" 1
2
x1 d" 2 x1, x2 e" 0
x1 max "
- x1x2 max
(x1 + 4)2 - 2 d" x2
x1 + x2 d" 1
x1 - x2 + 4 = 0
x1, x2 e" 0
x1 e" -10