1
2
-2023
-III
: : 100
: ,
-
1. () /
()
= x4 - 5xy3+ 6x2+ 2xz2- xyz.
2. (a) y = x3- 27x + 108 108
(b) : y = x3 - 27x + 108
-
3. () : .
() :
2
2
4. p = sqrt(9 + x) 7
-
5. (a) ?
(b) :
2z + 4y - z = 9
3x + y + 2z = 7
+ 3y - 3z = 4
6. (a) ?
(b) : 2 4 -1
3 1 2
1 3 -3
-
7. () -
() :
0.4
0.1
50
0.7
0.6
100
5
2
3
2
:
(i)
(ii)
(iii) 10 ,
8. () ? ...
?
() :
Z = 2x1 - 3x2
:
4
2
-2023
-III
: : 100
: ,
-
1. () /
()
= x4 - 5xy3+ 6x2+ 2xz2- xyz.
: A. /
( )
,
:
1. :
- (, f(x)) -
(, f(x,y,z))
2.
:
:
a) - :
• : f'(x)
• x : f'(x) = 0
• ( )
5
2
b) - :
•
•
• ( )
3. : (
, )
a) - :
• : f''(x)
• :
o f''(x) > 0,
o f''(x) < 0,
o f''(x) = 0, ( )
b) - :
• ( )
• :
o det(H) > 0 f_xx > 0,
o det(H) > 0 f_xx < 0,
o det(H) < 0,
o det(H) = 0, ( )
4.
:
,
5. :
6.
:
[-2, 4] f(x) = x^3 - 3x^2 - 9x + 5
6
2
1: f(x) = x^3 - 3x^2 - 9x + 5
: f'(x) = 3x^2 - 6x - 9 f'(x) = 0 : 3x^2 - 6x - 9 = 0 x^2 - 2x -
3 = 0 (x - 3)(x + 1) = 0 x = 3 x = -1
3: f''(x) = 6x - 6 x = 3 : f''(3) = 12 > 0, x = 3
x = -1 : f''(-1) = -12 < 0, x = -1
4: f(-2) = -8 - 12 + 18 + 5 = 3 f(4) = 64 - 48 - 36 + 5 = -15
5: f(-1) = -1 + 3 + 9 + 5 = 16 ( ) f(3) = 27
- 27 - 27 + 5 = -22 f(4) = -15 ( )
6: x = -1
16 x = 4 -15
- ,
B.
, - -
:
= x^4 - 5xy^3 + 6x^2 + 2xz^2 - xyz
- :
1. x (∂u/∂x): , y z
x
∂u/∂x = 4x^3 - 5y^3 + 12x + 2z^2 - yz
2. y (∂u/∂y): , x z y
∂u/∂y = -15xy^2 - xz
3. z (∂u/∂z): x y , z
∂u/∂z = 4xz - xy
- :
, - -
4. x (∂^2u/∂x^2): x ∂u/∂x :
∂^2u/∂x^2 = 12x^2 + 12
7
2
5. y (∂^2u/∂y^2): y ∂u/∂y :
∂^2u/∂y^2 = -30xy
6. z (∂^2u/∂z^2): z ∂u/∂z :
∂^2u/∂z^2 = 4x
7. x y (∂^2u/∂x∂y): ∂u/∂x y
( ∂u/∂y x ):
∂^2u/∂x∂y = -15y^2 - z
8. x z (∂^2u/∂x∂z): z ∂u/∂x
( x ∂u/∂z):
∂^2u/∂x∂z = 4z - y
9. y z (∂^2u/∂y∂z): ∂u/∂y z
( ∂u/∂z y ):
∂^2u/∂y∂z = -x
u
:
- :
1. ∂u/∂x = 4x^3 - 5y^3 + 12x + 2z^2 - yz x u
, y z x, y z
2. ∂u/∂y = -15xy^2 - xz y u , x z
x, y z
3. ∂u/∂z = 4xz - xy z u , x y
, x, y z
- :
4. ∂^2u/∂x^2 = 12x^2 + 12 x u (∂u/∂x) x
, x
5. ∂^2u/∂y^2 = -30xy y u (∂u/∂y) y
x y
6. ∂^2u/∂z^2 = 4x z u (∂u/∂z) z
x
8
2
7. ∂^2u/∂x∂y = -15y^2 - z y x
u ( ) u
x y
8. ∂^2u/∂x∂z = 4z - y z x u
( ), x z
9. ∂^2u/∂y∂z = -x z y u
( ), y z
u = x^4 - 5xy^3 + 6x^2 + 2xz^2 - xyz
:
1. -
2. - ( )
3.
4. x, y, z
, x, y,
z - u , ,
,
:
1. ( )
2. ,
3. ,
, , x, y,
z u x^4 y^3
- , xyz
9
2
,
, , ,
,
2. (a) y = x3- 27x + 108 108
(b) : y = x3 - 27x + 108
:(a) y = x³ - 27x + 108 108
:
1. y = x³ - 27x + 108
S-
2. : ,
3. :
,
:
1:
,
y = x³ - 27x + 108 y' = 3x² - 27
2:
, y' = 0 x :
3x² - 27 = 0 3x² = 27 x² = 9 x = ±3
x = 3 x = -3
3:
y' = 3x² - 27 y'' = 6x
10
2
4: x = 3 : y'' = 6(3) = 18 (,
) x = -3 : y'' = 6(-3) = -18 (, )
5 y- x = 3 ( ) : y = 3³ - 27(3) + 108 = 27
- 81 + 108 = 54
x = -3 ( ) : y = (-3)³ - 27(-3) + 108 = -27 + 81 + 108 = 162
6: : 162 : 54
: 162 - 54 = 108
, 108
(b) y = x³ - 27x + 108
, ()
:
: 162 (x = -3 ) : 54 (x = 3 )
, :
: y = x³ - 27x + 108
x³ x ( ) ,
(-27x) x = 0
(- x
), , , x
S-
, y = x³ -27x x = 0
, +108
: (x = 3 x = -3)
- ,
x = -3 , -3 -3
x- , y- 162 ""
""
x = 3 , "" -3 3 x-
, y- 54 , 162
:
, y' = 3x² - 27,
(
11
2
) , "" -
, y'' = 6x, (x >
0 ), (x <
0 ),
( x = 0)
: x = 3
x = -3 , x = 0 (x³ x) -
- ( 108)
: , -
,
( )
, x (
), y ( ),
:
• 3 -3 (
)
• -3 , 162
• 3 , 54
• -3 3 54 162
: ,
y = x³ - 27x + 108 ,
:
•
( x )
• x = -3, y = 162
• x = 3, y = 54
• - x ,
"108 " : 108 ,
12
2
108 y = ax³ + bx + c
2c
: : 162 : 54 : 162 - 54 = 108
: 108 2 * 108 = 216
: ,
:
1. Y- x = 0, y = 108. y-
2. x-: y = 0 , : x³
- 27x + 108 = 0
3.
: y'' = 6x,
x = 0 ,
: , y = ax³ +
bx + c :
1.
2.
3.
4.
,
: y = x³ - 27x + 108 ,
,
,
, , S-
,
, ,
,
13
2
- -
,
-
3. () : .
() :
: () : ∫x^n log(x) dx
x^n log(x) ,
:
∫u dv = uv - ∫v du
u v, x
:
1. , u dv : u = log(x) dv = x^n dx
2. du v : du = (1/x) dx ( log(x) ) v =
(x^(n+1))/(n+1) ( x^n )
3. : ∫x^n log(x) dx = log(x) * (x^(n+1))/(n+1) -
∫(x^(n+1))/(n+1) * (1/x) dx
4. : ∫x^n log(x) dx = log(x) * (x^(n+1))/(n+1) - (1/(n+1)) ∫x^n dx
5. : ∫x^n dx = (x^(n+1))/(n+1)
6. : ∫x^n log(x) dx = log(x) * (x^(n+1))/(n+1) - (1/(n+1)) *
(x^(n+1))/(n+1)
7. : ∫x^n log(x) dx = (x^(n+1)/(n+1)) * (log(x) - 1/(n+1)) + C
C
()
:
14
2
• x^(n+1)/(n+1) x^n
• (x) (x)
• -1/(n+1)
• C
n = -1 n ( n = -1 ,
)
, ()
() : ∫log/(1+log)^2 dx
-- :
1. , u = log(x)
, du = (1/x) dx x = e^u ( e^(log(x)) = x)
2. u : ∫log/(1+log)^2 dx = ∫u/(1+u)^2 * e^u du
3. :
∫u/(1+u)^2 * e^u du
4.
v = u/(1+u)^2 dw = e^u du
5. dv w : dv = ((1+u)^2 - 2u(1+u)) / (1+u)^4 du = (1-u) / (1+u)^3 du
w = e^u
6. : ∫u/(1+u)^2 * e^u du = u/(1+u)^2 * e^u - ∫e^u * (1-u) /
(1+u)^3 du
7. I : I = ∫e^u * (1-u)
/ (1+u)^3 du
8. : I = ∫e^u / (1+u)^3 du - ∫u*e^u / (1+u)^3
du
9. : ∫e^u / (1+u)^3 du v = 1+u dv = du u = v-
1 : ∫e^(v-1)
/ v^3 dv = (1/e) * ∫e^v / v^3 dv
10. : (1/e) * (-e^v / (2v^2) - e^v / (2v) + 1/2 * e^v *
ln|v| + C)
15
2
11. u = v-1: (-e^u / (2(1+u)^2) - e^u / (2(1+u)) + 1/2 * e^u *
ln|1+u| + C)
12. : -∫u*e^u / (1+u)^3 du ,
u
13. (
), : ∫log/(1+log)^2 dx = x / (1+log(x)) + ln|1+log(x)| + C
()
:
• x / (1+log(x))
• ln|1+log(x)|
• , C
, (), :
1.
2.
3.
4.
5.
, :
•
•
•
•
, ,
() ,
,
,
:
1. : ,
16
2
2. : ,
3. : ,
,
4. : , --
,
5. : ,
,
,
-
:
1.
2.
3. ,
4. -,
5.
, - ,
-
, -
-
17
2
4. p = sqrt(9 + x) 7
: ,
:
,
,
, ,
, -- :
:
1. : p = sqrt(9 + x) p x
2. : 7
1:
, (7 )
:
= sqrt(9 + x) = sqrt(9 + 7) = sqrt(16) = 4
4 ( )
:
, (p = 4) (p =
sqrt(9 + x)) , x = 0 x = 7
, y = 4 (
) ,
3:
,
= × = 4 × 7 = 28
18
2
= ∫[0 7] sqrt(9 + x) dx
= -
4:
, 0 7 sqrt(9 + x)
, :
∫ sqrt( + x) dx
, : u = 9 + x du/dx = 1 dx =
du
: ∫ sqrt(u) du = (2/3)u^(3/2) + C
, x = u - 9 : (2/3)(9 + x)^(3/2) + C
0 7 :
[(2/3)(9 + 7)^(3/2)] - [(2/3)(9 + 0)^(3/2)] = (2/3)(16)^(3/ 2) - (2/3)(9)^(3/2) = (2/3) × 64 - (2/3)
× 27 = 42.67 - 18 = 24.67
5:
= - = 28 - 24.67 =
3.33
, 3.33 ( $3.33)
, :
1. : 4 , 7
28
2. : p = sqrt(9 + x)
:
o : sqrt(9 + 1) = sqrt(10) ≈ 3.16
o 7 : sqrt(9 + 7) = sqrt(16) = 4
3. : 4
, 3.16
4. : ( 4 )
( 3.16 )
19
2
5. : 7 ,
$3.33
6. : $3.33 ""
:
• - (
, )
• 4
• 3.16
• , 4
• 7 4 ,
,
• $4
$3.33
:
1. :
2. :
3.
,
4. :
(
)
5. :
, -
:
20
2
1. 5 :
o 5 × 7 = 35
o ,
2. $4 10 :
o x = 10
o
3.
4. p = sqrt(4 + x) :
o
o ,
, $3.33 , $4
,
,
, , ,
, ,
,
, !
21
2
-
5. (a) ?
(b) :
2z + 4y - z = 9
3x + y + 2z = 7
+ 3y - 3z = 4
: ():
?
,
"det(A)" , A
, 2x2 :
|| ||
: ad - bc
3x3 :
|abc| |def| |ghi|
: a(ei-fh) - b(di-fg) + c(dh-eg)
,
:
1. :
2. 1x1 : [a] , 'a'
3. :
,
4. :
k ,
k
22
2
5. ( ) ( )
:
6. , :
,
7. , :
8. ( ) :
9. : det(AB) = det(A) *
det(B)
10. : det(A^(-
1)) = 1 / det(A)
11. : det(A^T) =
det(A)
12. ( ), :
, :
• (
)
•
•
•
•
():
, :
2x + 4y - z = 9 3x + y + 2z = 7 x + 3y - 3z = 4
:
23
2
1:
AX = B , :
A = | 2 4 -1 | X = |x| B = | 9 | | 3 1 2 | |y| | 7 | | 1 3 -3 | |z| | 4 |
2: A ( D )
= | 2 4 -1 | | 3 1 2 | | 1 3 -3 |
= 2(1(-3)-2(3)) - 4(3(-3)-2(1)) + (-1)(3(3)-1(1)) = 2(-3-6) - 4(-9-2) + (-1)(9-1) = 2(-9) - 4(-11) +
(-8) = -18 + 44 - 8 = 18
3:A B Dx, Dy, Dz
= | - | | | | - |
= | 2 9 -1 | | 3 7 2 | | 1 4 -3 |
= | 2 4 9 | | 3 1 7 | | 1 3 4 |
4:Dx, Dy, Dz
= 9(1(-3)-2(3)) - 4(7(-3)-2(4)) + (-1)(7(3)-4(1)) = 9(-3-6) - 4(-21-8) + (-1)(21-4) = 9(-9) -
4(-29) + (-17) = -81 + 116 - 17 = 18
= 2(7(-3)-2(4)) - 9(3(-3)-2(1)) + (-1)(3(4)-1(7)) = 2(-21) -8) - 9(-9-2) + (-1)(12-7) = 2(-29) -
9(-11) + (-5) = -58 + 99 - 5 = 36
= 2(1(4)-3(7)) - 4(3(4)-1(7)) + 9(3(3)-1(1)) = 2(4-21) - 4( 12-7) + 9(9-1) = 2(-17) - 4(5) +
9(8) = -34 - 20 + 72 = 18
5:x, y, z
: x = Dx / D y = Dy / D z = Dz / D
: x = 18 / 18 = 1 y = 36 / 18 = 2 z = 18 / 18 = 1
, : x = 1, y = 2, z = 1
:
1. 2x + 4y - z = 9 2(1) + 4(2) - 1 = 9 2 + 8 - 1 = 9 9 = 9 ✓
2. 3x + y + 2z = 7 3(1) + 2 + 2(1) = 7 3 + 2 + 2 = 7 7 = 7 ✓
3. x + 3y - 3z = 4 1 + 3(2) - 3(1) = 4 1 + 6 - 3 = 4 4 = 4 ✓
,
24
2
:
-
:
1. D " " D = 0 ,
2. Dx, Dy, Dz (B)
3. Dx, Dy, Dz D , A
"" , x, y z
:
1. ,
2.
3.
:
1. (3 4 )
2.
3.
( D = 0 )
,
,
,
:
,
( -
)
, ,
, ,
25
2
,
- ,
,
6. (a) ?
(b) : 2 4 -1
3 1 2
1 3 -3
: :
,
, :
1. ? ,
:
[2 3 1] [4 1 -2] [0 2 5]
3x3 (3 3 )
2. ?
:
• 1
• 2
• 3 -
26
2
3. ?
:
1: 3x3 :
[1 2 3] [2 4 6] [3 6 9]
, 3x3 , 3 ,
:
•
•
-
,
3x3 1
2: 3x3 :
[1 2 3] [0 1 4] [5 6 7]
:
•
•
•
, 3
4. ?
:
• ,
• ,
• ,
• ,
27
2
5. ,
-
, :
[1 2 3] [2 4 6] → [1 2 3] → [1 2 3] [3 6 9] [0 0 0] [0 0 0] [0 0 0] [0 0 0]
- , 1
:
, :
, ,
; (
) ,
:
[2 4 -1] [3 1 2] [1 3 -3]
:
1.
2.
3.
1:
3x3 , :
det(A) = a(ei - fh) - b(di - fg) + c(dh - )
: [abc] [def] [ghi]
: a = 2, b = 4, c = -1 d = 3, e = 1, f = 2 g = 1, h = 3, i = -3
det() = 2[(1)(-3) - (2)(3)] - 4[(3)(-3) - (2)(1)] + (-1)[(3)( 3) - (1)(1)] = 2(-3 - 6) - 4(-9 - 2) + (-1)(9
- 1) = 2(-9) - 4(-11) + ( -1)(8) = -18 + 44 - 8 = 18
,
2:
:
a. b.
c.
28
2
: 11 = +(1*-3 - 23) = -9
12 = -(3-3 - 21) = +11
13 = +(33 - 11) = +8 21 = -(4-3 - (-1)3) = +15 22 = +(2-3 - (-1)1) = -5 23 = -(23 - 41) = -
2 31 = +(42 - (-1)1) = +9 32 = -(22 - (-1)3) = -7 33 = +(21 - 4*3) = -10
:
[-9 11 8 ] [15 -5 -2 ] [9 -7 -10]
3:
= (1/18) *
[(-9/18) (11/18) (8/18) ] [(15/18) (-5/18) (-2/18)] [(9/18) (-7/18) ( -10/18)]
:
[-1/2 11/18 4/9 ] [5/6 -5/18 -1/9 ] [1/2 -7/18 -5/9 ]
, :
[2 4 -1] [-1/2 11/18 4/9 ] [3 1 2] * [5/6 -5/18 -1/9 ] [1 3 -3] [1/2 -7/ 18 -5/9 ]
[1 0 0] [0 1 0] [0 0 1]
1. ? A , A^(-1) ,
A :
* ^(-1) = ^(-1) * =
I .
2.
•
• ( "" "-"
)
• ,
• (^(-1))^(-1) =
• (AB)^(-1) = B^(-1)A^(-1) ( A B )
29
2
3. :
•
•
• -
•
•
4. ,
:
•
•
•
,
:
(
)
, 3x3 , 3
( )
:
1. :,
2. : () ,
3. - ,
4. -
30
2
5. ,
6. : ,
7. : ,
,
" " ,
"" ,
,
, ,
,
, - ,
,
-
7. () -
() :
0.4
0.1
50
0.7
0.6
100
5
2
:
(i)
31
2
(ii)
(iii) 10 ,
: -
- 1930
- :
) :-
,
, -
,
) : ,
,
) -
,
)
) -
- ,
,
) : , -
(
CO2 ) ,
)
) :
-
2.
32
2
,
: : 0.4 0.1 50 0.7
0.6 100 5 2
:(i) (ii) (iii) ,
10
1:
-
:
• 1 0.4 , 0.7 5
• 1 0.1 , 0.6 2
2:
'S' 'C'
: S = 0.4S + 0.1C + 50 ( ) C = 0.7S + 0.6C + 100 ( )
: 0.6S - 0.1C = 50 -0.7S + 0.4C = 100
:
7 6 : 4.2S - 0.7C = 350 -4.2S + 2.4C = 600
: 1.7C = 950 C = 950 / 1.7 = 558.82
C : S = 0.4S + 0.1(558.82) + 50 0.6S =
105.882 S = 176.47
, : : 176.47 : 558.82
3:
,
:
= 5 * 176.47 = 882.35 = 2 * 558.82 = 1117.64
33
2
= 882.35 + 1117.64 = 2000
4:
,
'Ps' 'Pc' 10
• : Ps = 0.4Ps + 0.7Pc + 5 * 10 Pc = 0.1Ps + 0.6Pc + 2 * 10
• : 0.6Ps - 0.7Pc = 50 -0.1Ps + 0.4Pc = 20
• 4 6 : 2.4Ps - 2.8Pc = 200 -0.6Ps + 2.4Pc =
120
• : 1.8Ps - 0.4Pc = 320
• : 0.6(320 + 0.4Pc)/1.8 - 0.7Pc = 50 106.67
+ 0.13Pc - 0.7Pc = 50 106.67 - 0.57Pc = 50 -0.57Pc = -56.67 Pc = 99.42
• Ps : 0.6Ps - 0.7(99.42) = 50 0.6Ps = 119.59 Ps =
199.32
• , : : 199.32
: 99.42
:
1. : (
) ,
176.47 558.82
( 50 100 )
2. 2000
3. :
, (
199.32 99.42 ),
:
1. :
-
34
2
2. :
3. :
,
4. :
,
, ,
5.
, ,
6. :
:
- ,
:
1. : ,
2. : ,
3. : ,
4.
5.
:
- ,
, -
35
2
, ,
,
, ,
, -
,
8. () ? ...
?
() :
Z = 2x1 - 3x2
:
:
" "
(≥)
: ... ,
:
1. :
,
, ,
36
2
2. :
3. : ,
,
4. : ,
,
,
5. ,
,
,
,
2.
, :
Z = 2x₁ - 3x₂ : 4x₁ + 5x₂ ≤ 40 x₁ + 3x₂ ≤ 12 x₁ - x₂ ≥ 2 x₁ ≥ 4 x₁, x₂ ≥
0
1:
:
1. 4x₁ + 5x₂ = 40
2. x₁ + 3x₂ = 12
3. x₁ - x₂ = 2
4. x₁ = 4
5. x₂ = 0 (x-)
6. x₁ = 0 (y-)
, :
1. 4x₁ + 5x₂ = 40 x₁-: (10, 0) x₂-: (0, 8)
2. x₁ + 3x₂ = 12 x₁-: (12, 0) x₂-: (0, 4)
3. x₁ - x₂ = 2 x₁-: (2, 0) x₂-: (0, -2)
37
2
4. x₁ = 4 x₁ = 4
2:
,
3:
:
A. x₁ = 4 x₁ + 3x₂ = 12 4 + 3x₂ = 12 3x₂ = 8 x₂ = 8/3 A: (4, 8/3)
. x₁ = 4 4x₁ + 5x₂ = 40 4(4) + 5x₂ = 40 16 + 5x₂ = 40 5x₂ = 24 x₂ = 24/5 B: (4,
24/5)
. x₁ - x₂ = 2 x₁ + 3x₂ = 12 x₁ = x₂ + 2 x₁ + 3x₂ = 12 : (x₂ +
2) + 3x₂ = 12 4x₂ + 2 = 12 4x₂ = 10 x₂ = 5/2 x₁ = 5/2 + 2 = 9/2 C: (9/2, 5/2)
. x₁ - x₂ = 2 4x₁ + 5x₂ = 40 4x₁ + 5x₂ = 40 x₁ = x₂ + 2 : 4(x₂ +
2) + 5x₂ = 40 4x₂ + 8 + 5x₂ = 40 9x₂ = 32 x₂ = 32/9 x₁ = 32/9 + 2 = 50/9 D: (50/9, 32/9)
4:
= 2x₁ - 3x₂
• A (4, 8/3) : Z = 2(4) - 3(8/3) = 8 - 8 = 0
• B (4, 24/5): Z = 2(4) - 3(24/5) = 8 - 14.4 = -6.4
• C (9/2, 5/2): Z = 2(9/2) - 3(5/2) = 9 - 7.5 = 1.5
• D (50/9, 32/9): Z = 2(50/9) - 3(32/9) = 100/9 - 96/9 = 4/9 ≈ 0.44
5:
Z , C (9/2, 5/2) ,
Z = 1.5
, : x₁ = 9/2 = 4.5 x₂ = 5/2 = 2.5 Z = 1.5
3.
, :
(i) : Z = 2x₁ - 3x₂
, Z = 1.5
( , $1,500 $1,000 )
38
2
(ii) : ,
, :
:
• x₁ 2
• x₂ 3
, : = 2x₁ + 3x₂ = 2(4.5) + 3(2.5) = 9 + 7.5 = 16.5
(iii) , LPP
,
, :
1. ( y₁, y₂, y₃ y₄ )
2. ,
: 4y₁ + y₂ + y₃ + y₄ ≥ 2 (x₁ ) 5y₁ + 3y₂ - y₃ ≥ -3
(x₂ )
3. 10
4. ,
5.
6. :
:
o x₁ - x₂ ≥ 2 ( x₁ - x₂ = 2)
o x₁ + 3x₂ ≤ 12 (4.5 + 3(2.5) = 12)
7. : ( )
, x₁ + 3x₂ ≤ 12 ,
?
39
2
8. :
9. : 1 (x₁) 4.5 2 (x₂) 2.5
2
,
10. : (-3) x₂
x₂
, ,
x₂
11. : ( C D )
,
12. : 4x₁ + 5x₂ ≤ 40 (4(4.5) + 5(2.5) = 30.5 < 40)
,
13. : x₁ ≥ 4 (x₁ = 4.5 > 4)
1 ,
,
14. : ,
( ), ( )
, ,
( )
15. ,
- -
, ,
40
2
16. - :x₁, x₂ ≥ 0
,
" "
17. : x₁ x₂
- x₂
,
18. x₁ - x₂ = 2 1 2
, 1
2 ,
19. : ,
( )
, x₁ + 3x₂ ≤ 12
20. : (1.5)
21. x₂
,
-
22. :
, 4x₁ + 5x₂ ≤ 40 , (4(4.5) + 5(2.5)) / 40 = 30.5 / 40 =
76.25%
23. :
-
(
)
24. : ,
?
41
2
25.
,
,
, ,
,
: ( )
,
