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GTLN-GTNN V B畉T 畉NG TH畛C TRONG 畛 THI 畉I H畛C T畛 2002 畉N 2013
Bi 1 (H A2003) Cho x ,y ,z l ba s畛 d動董ng v x + y + z 1 . Ch畛ng minh r畉ng
1
1
1
+ y 2 + 2 + z 2 + 2 82
2
x
y
z
1
S : x = y = z =
3
Bi 2 (H B2003) T狸m gi叩 tr畛 nh畛 nh畉t c畛a hm s畛 :
x2 +
y = x + 4 x2
S : Maxy = y (2) = 2 2 ; Miny = y (2) = 2
[ 2;2]
[ 2;2]
Bi 3 (H D2003) T狸m gi叩 tr畛 l畛n nh畉t v nh畛 nh畉t c畛a hm s畛 tr棚n o畉n [-1; 2].
x +1
y=
x2 + 1
S : Maxy = y (1) = 2 ; Miny = y (1) = 0
[ 1;2]
[ 1;2]
3
Bi 4 (H B2004) T狸m gi叩 tr畛 l畛n nh畉t v nh畛 nh畉t c畛a hm s畛 tr棚n o畉n 錚1; e 錚 .
錚
錚
2
ln x
y=
x
4 Miny = y (1) = 0
2
S : Maxy = y (e ) = 2 ; 錚1;e3 錚
e
錚1;e3 錚
錚
錚
錚
錚
Bi 5 (H A2005) Cho x, y, z l c叩c s畛 d動董ng th畛a m達n
1 1 1
+ + = 4 . Ch畛ng minh r畉ng
x y z
1
1
1
+
+
1
2x + y + z x + 2 y + z x + y + 2z
3
S : x = y = z =
4
Bi 6 (H B2005) Ch畛ng minh r畉ng v畛i m畛i x R , ta c坦 .
x
x
x
錚 12 錚 錚 15 錚 錚 20 錚
x
x
x
錚 歎 + 錚 歎 + 錚 歎 3 + 4 + 5 . Khi no 畉ng th畛c x畉y ra?
錚 5錚 錚 4錚 錚 3 錚
S : x = 0
Bi 7 (H D2005) Cho c叩c s畛 d動董ng x, y, z th畛a m達n xyz = 1. Ch畛ng minh r畉ng :
1 + x3 + y 3
1 + y3 + z3
1 + z 3 + x3
+
+
3 3
xy
yz
zx
.Khi no 畉ng th畛c x畉y ra?
S : x = y = z = 1
Bi 8 (H A2006) Cho hai s畛 th畛c thay 畛i v th畛a m達n i畛u ki畛n: ( x + y ) xy = x 2 + y 2 xy .
1 1
T狸m gi叩 tr畛 l畛n nh畉t c畛a bi畛u th畛c A = 3 + 3 .
x
y
1
S : MaxA = 16 x = y =
2
Bi 9 (H B2006) Cho x,y l c叩c s畛 th畛c thay 畛i. T狸m gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c:
A = ( x 1) 2 + y 2 + ( x + 1) 2 + y 2 + | y 2 |
1
S : MinA = 2 + 3 x = 0; y =
3
Bi 10 (H A2007) Cho x , y , z l c叩c s畛 th畛c d動董ng thay 畛i v th畛a m達n i畛u ki畛n xyz = 1 . T狸m gi叩
C担 s担短 boi d旦担探ng va棚n ho湛a- luye辰n thi 単a誰i ho誰c Cao Nguye但n-128/39 YwangBMT
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tr畛 nh畛 nh畉t c畛a bi畛u th畛c P =
x2 ( y + z)
y 2 ( z + x)
z 2 ( x + y)
+
+
y y + 2z z z z + 2x x x x + 2 y y
S : MinP = 2 x = y = z = 1
Bi 11 (H B2007) Cho x , y , z l ba s畛 th畛c d動董ng thay 畛i . T狸m gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c :
x 1
y 1
z 1
P = x( + ) + y( + ) + z ( + )
2 yz
2 zx
2 xy
9
S : MinP = x = y = z = 1
2
Bi 12 (H D2007) Cho a b > 0 . Ch畛ng minh r畉ng :
b
a
錚 a 1 錚 錚 b 1錚
錚2 + a 歎 錚2 + b 歎
2 錚 錚
2 錚
錚
Bi 13 (H B2008) Cho hai s畛 th畛c x, y thay 畛i v th畛a m達n h畛 th畛c x2 + y2 =1. T狸m gi叩 tr畛 l畛n nh畉t
2( x 2 + 6 xy )
v gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c P =
.
1 + 2 xy + 2 y 2
3
1
3
2
錚
錚
錚 x = 10 ; y = 10
錚 x = 13 ; y = 13
S : MaxP = 3 錚
; MinP = 6 錚
3
1
3
2
錚
錚
錚 x = 10 ; y = 10
錚 x = 13 ; y = 13
錚
錚
Bi 14 (H D2008) Cho x,y l hai s畛 th畛c kh担ng 但m thay 畛i. T狸m gi叩 tr畛 l畛n nh畉t v gi叩 tr畛 nh畛 nh畉t
( x y )(1 xy )
c畛a bi畛u th畛c : P =
.
(1 + x ) 2 (1 + y ) 2
1
1
S : MaxP = x = 1; y = 0; MinP = x = 0; y = 1
4
4
Bi 15 (H A2009) Ch畛ng minh r畉ng v畛i m畛i s畛 th畛c d動董ng x, y, z tho畉 m達n x(x + y + z)=3yz,
ta c坦: (x + y)3 + (x + z)3 + 3(x + y)(x + z)(y + z) 5(y + z)3
S : x = y = z
Bi 16 (H B2009) Cho c叩c s畛 th畛c x, y thay 畛i v tho畉 m達n (x + y) 3 + 4xy 2. T狸m gi叩 tr畛 nh畛 nh畉t
c畛a bi畛u th畛c :A = 3(x4 + y4 + x2y2) 2(x2 + y2) + 1
S : MinA =
9
1
x= y=
16
2
Bi 17 (H D2009) Cho c叩c s畛 th畛c kh担ng 但m x, y thay 畛i v th畛a m達n x + y = 1. T狸m gi叩 tr畛 l畛n nh畉t
v gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c S = (4x2 + 3y)(4y2 + 3x) + 25xy.
錚
錚
2+ 3
2 3
錚x =
錚x =
25
1
191
錚
錚
4
4
x = y = ; MinP =
錚
S : MaxS =
ho畉c 錚
2
2
16
錚y = 2 3
錚y = 2+ 3
錚
錚
錚
4
錚
4
Bi 18 (H B2010) Cho c叩c s畛 th畛c a ,b ,c kh担ng 但m th畛a m達n a + b + c = 1 . T狸m gi叩 tr畛 nh畛 nh畉t c畛a
bi畛u th畛c M = 3(a 2b 2 + b 2 c 2 + c 2 a 2 ) + 3(ab + bc + ca) + 2 a 2 + b 2 + c 2
S : MinM = 2 ( a, b, c) l m畛t trong c叩c b畛 s畛 : (1;0;0), (0;1;0), (0;0;1)
Bi 20 (H D2010) T狸m gi叩 tr畛 nh畛 nh畉t c畛a hm s畛
y = x 2 + 4 x + 21 x 2 + 3x + 10
1
S : Miny = 2 x =
3
Bi 21 (H A2011) Cho x, y, z l ba s畛 th畛c thu畛c o畉n [1; 4] v x y, x z. T狸m gi叩 tr畛 nh畛 nh畉t c畛a bi畛u
C担 s担短 boi d旦担探ng va棚n ho湛a- luye辰n thi 単a誰i ho誰c Cao Nguye但n-128/39 YwangBMT
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bi畛u th畛c P =
x
y
z
+
+
2x + 3y y + z z + x
34
x = 4; y = 1; z = 2
33
Bi 22 (H B2011) Cho c叩c s畛 th畛c a, b, l c叩c s畛 th畛c d動董ng th畛a m達n i畛u ki畛n :
錚 a 3 b3 錚 錚 a 2 b 2 錚
2(a 2 + b 2 ) + ab = (a + b)(ab + 2) . T狸m gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c P = 4 錚 3 + 3 歎 9 錚 2 + 2 歎.
a 錚
錚b a 錚 錚b
錚a = 2
錚a = 1
23
S : MinP = 錚
ho畉c 錚
4
錚b = 1
錚b = 2
S : MinP =
Bi 23 (H D2011NC) T狸m gi叩 tr畛 l畛n nh畉t v nh畛 nh畉t c畛a hm s畛 tr棚n o畉n [ 0; 2] .
y=
2 x 2 + 3x + 3
x +1
17
S : Miny = y (0) = 3 ; Maxy = y (2) =
[ 0;2]
3
[ 0;2]
Bi 24 (H A2012) Cho c叩c s畛 th畛c x, y, z th畛a m達n i畛u ki畛n x +y + z = 0. T狸m gi叩 tr畛 nh畛 nh畉t c畛a
bi畛u th畛c P = 3 x y + 3 y z + 3 z x 6 x 2 + 6 y 2 + 6 z 2 .
S : MinP = 3 x = y = z = 0
Bi 25 (H B2012) Cho c叩c s畛 th畛c x, y, z th畛a m達n c叩c i畛u ki畛n x + y + z = 0 v x 2 + y 2 + z 2 = 1.
T狸m gi叩 tr畛 l畛n nh畉t c畛a bi畛u th畛c P = x 5 + y 5 + z 5 .
5 6
6
6
x=
;y=z=
36
3
6
Bi 26 (H D2012) Cho c叩c s畛 th畛c x, y th畛a m達n (x 4)2 + (y 4)2 + 2xy 32. T狸m gi叩 tr畛 nh畛 nh畉t
c畛a bi畛u th畛c A = x3 + y3 + 3(xy 1)(x + y 2).
17 5 5
1+ 5
S : MinA =
x= y=
4
4
Bi 27 (H A2013) Cho c叩c s畛 th畛c d動董ng a, b, c th畛a m達n i畛u ki畛n (a + c)(b + c) = 4c 2 . T狸m gi叩 tr畛
S : MaxP =
nh畛 nh畉t c畛a bi畛u th畛c P =
32a 3
32b3
a 2 + b2
+
(b + 3c)3 (a + 3c)3
c
S : MinP = 1 2 x = y = 1
Bi 28 (H B2013) Cho a, b, c l c叩c s畛 th畛c d動董ng . T狸m gi叩 tr畛 l畛n nh畉t c畛a bi畛u th畛c :
4
9
P=
a 2 + b 2 + c2 + 4 (a + b) (a + 2c)(b + 2c)
5
S : MaxP = a = b = c = 2
8
Bi 29 (H D2013) Cho x, y l c叩c s畛 th畛c d動董ng th畛a m達n i畛u ki畛n xy y 1 . T狸m gi叩 tr畛 l畛n nh畉t
x+y
x 2y
c畛a bi畛u th畛c: P =
2
2
6(x + y)
x xy + 3y
5 7
1
+
x= ;y=2
3 30
2
Bi 30 (H D2013NC) T狸m gi叩 tr畛 l畛n nh畉t v gi叩 tr畛 nh畛 nh畉t c畛a hm s畛 tr棚n o畉n [ 0; 2] .
S : MaxP =
2 x 2 3x + 3
x +1
S : Minf(x) = f (1) = 1 ; Maxf(x) = f (0) = 3
[ 0;2]
[ 0;2]
f ( x) =
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Gtln gtnn va bdt 2002 -2013
- 1. C担 s担短 BDVH - LTH Cao Nguye但nBMT www.luyenthicaonguyen.com C: 128/39 Ywang - BMT T: 0984.959.465-0945.46.00.44 GTLN-GTNN V B畉T 畉NG TH畛C TRONG 畛 THI 畉I H畛C T畛 2002 畉N 2013 Bi 1 (H A2003) Cho x ,y ,z l ba s畛 d動董ng v x + y + z 1 . Ch畛ng minh r畉ng 1 1 1 + y 2 + 2 + z 2 + 2 82 2 x y z 1 S : x = y = z = 3 Bi 2 (H B2003) T狸m gi叩 tr畛 nh畛 nh畉t c畛a hm s畛 : x2 + y = x + 4 x2 S : Maxy = y (2) = 2 2 ; Miny = y (2) = 2 [ 2;2] [ 2;2] Bi 3 (H D2003) T狸m gi叩 tr畛 l畛n nh畉t v nh畛 nh畉t c畛a hm s畛 tr棚n o畉n [-1; 2]. x +1 y= x2 + 1 S : Maxy = y (1) = 2 ; Miny = y (1) = 0 [ 1;2] [ 1;2] 3 Bi 4 (H B2004) T狸m gi叩 tr畛 l畛n nh畉t v nh畛 nh畉t c畛a hm s畛 tr棚n o畉n 錚1; e 錚 . 錚 錚 2 ln x y= x 4 Miny = y (1) = 0 2 S : Maxy = y (e ) = 2 ; 錚1;e3 錚 e 錚1;e3 錚 錚 錚 錚 錚 Bi 5 (H A2005) Cho x, y, z l c叩c s畛 d動董ng th畛a m達n 1 1 1 + + = 4 . Ch畛ng minh r畉ng x y z 1 1 1 + + 1 2x + y + z x + 2 y + z x + y + 2z 3 S : x = y = z = 4 Bi 6 (H B2005) Ch畛ng minh r畉ng v畛i m畛i x R , ta c坦 . x x x 錚 12 錚 錚 15 錚 錚 20 錚 x x x 錚 歎 + 錚 歎 + 錚 歎 3 + 4 + 5 . Khi no 畉ng th畛c x畉y ra? 錚 5錚 錚 4錚 錚 3 錚 S : x = 0 Bi 7 (H D2005) Cho c叩c s畛 d動董ng x, y, z th畛a m達n xyz = 1. Ch畛ng minh r畉ng : 1 + x3 + y 3 1 + y3 + z3 1 + z 3 + x3 + + 3 3 xy yz zx .Khi no 畉ng th畛c x畉y ra? S : x = y = z = 1 Bi 8 (H A2006) Cho hai s畛 th畛c thay 畛i v th畛a m達n i畛u ki畛n: ( x + y ) xy = x 2 + y 2 xy . 1 1 T狸m gi叩 tr畛 l畛n nh畉t c畛a bi畛u th畛c A = 3 + 3 . x y 1 S : MaxA = 16 x = y = 2 Bi 9 (H B2006) Cho x,y l c叩c s畛 th畛c thay 畛i. T狸m gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c: A = ( x 1) 2 + y 2 + ( x + 1) 2 + y 2 + | y 2 | 1 S : MinA = 2 + 3 x = 0; y = 3 Bi 10 (H A2007) Cho x , y , z l c叩c s畛 th畛c d動董ng thay 畛i v th畛a m達n i畛u ki畛n xyz = 1 . T狸m gi叩 C担 s担短 boi d旦担探ng va棚n ho湛a- luye辰n thi 単a誰i ho誰c Cao Nguye但n-128/39 YwangBMT Trang 1
- 2. C担 s担短 BDVH - LTH Cao Nguye但nBMT www.luyenthicaonguyen.com C: 128/39 Ywang - BMT T: 0984.959.465-0945.46.00.44 tr畛 nh畛 nh畉t c畛a bi畛u th畛c P = x2 ( y + z) y 2 ( z + x) z 2 ( x + y) + + y y + 2z z z z + 2x x x x + 2 y y S : MinP = 2 x = y = z = 1 Bi 11 (H B2007) Cho x , y , z l ba s畛 th畛c d動董ng thay 畛i . T狸m gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c : x 1 y 1 z 1 P = x( + ) + y( + ) + z ( + ) 2 yz 2 zx 2 xy 9 S : MinP = x = y = z = 1 2 Bi 12 (H D2007) Cho a b > 0 . Ch畛ng minh r畉ng : b a 錚 a 1 錚 錚 b 1錚 錚2 + a 歎 錚2 + b 歎 2 錚 錚 2 錚 錚 Bi 13 (H B2008) Cho hai s畛 th畛c x, y thay 畛i v th畛a m達n h畛 th畛c x2 + y2 =1. T狸m gi叩 tr畛 l畛n nh畉t 2( x 2 + 6 xy ) v gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c P = . 1 + 2 xy + 2 y 2 3 1 3 2 錚 錚 錚 x = 10 ; y = 10 錚 x = 13 ; y = 13 S : MaxP = 3 錚 ; MinP = 6 錚 3 1 3 2 錚 錚 錚 x = 10 ; y = 10 錚 x = 13 ; y = 13 錚 錚 Bi 14 (H D2008) Cho x,y l hai s畛 th畛c kh担ng 但m thay 畛i. T狸m gi叩 tr畛 l畛n nh畉t v gi叩 tr畛 nh畛 nh畉t ( x y )(1 xy ) c畛a bi畛u th畛c : P = . (1 + x ) 2 (1 + y ) 2 1 1 S : MaxP = x = 1; y = 0; MinP = x = 0; y = 1 4 4 Bi 15 (H A2009) Ch畛ng minh r畉ng v畛i m畛i s畛 th畛c d動董ng x, y, z tho畉 m達n x(x + y + z)=3yz, ta c坦: (x + y)3 + (x + z)3 + 3(x + y)(x + z)(y + z) 5(y + z)3 S : x = y = z Bi 16 (H B2009) Cho c叩c s畛 th畛c x, y thay 畛i v tho畉 m達n (x + y) 3 + 4xy 2. T狸m gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c :A = 3(x4 + y4 + x2y2) 2(x2 + y2) + 1 S : MinA = 9 1 x= y= 16 2 Bi 17 (H D2009) Cho c叩c s畛 th畛c kh担ng 但m x, y thay 畛i v th畛a m達n x + y = 1. T狸m gi叩 tr畛 l畛n nh畉t v gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c S = (4x2 + 3y)(4y2 + 3x) + 25xy. 錚 錚 2+ 3 2 3 錚x = 錚x = 25 1 191 錚 錚 4 4 x = y = ; MinP = 錚 S : MaxS = ho畉c 錚 2 2 16 錚y = 2 3 錚y = 2+ 3 錚 錚 錚 4 錚 4 Bi 18 (H B2010) Cho c叩c s畛 th畛c a ,b ,c kh担ng 但m th畛a m達n a + b + c = 1 . T狸m gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c M = 3(a 2b 2 + b 2 c 2 + c 2 a 2 ) + 3(ab + bc + ca) + 2 a 2 + b 2 + c 2 S : MinM = 2 ( a, b, c) l m畛t trong c叩c b畛 s畛 : (1;0;0), (0;1;0), (0;0;1) Bi 20 (H D2010) T狸m gi叩 tr畛 nh畛 nh畉t c畛a hm s畛 y = x 2 + 4 x + 21 x 2 + 3x + 10 1 S : Miny = 2 x = 3 Bi 21 (H A2011) Cho x, y, z l ba s畛 th畛c thu畛c o畉n [1; 4] v x y, x z. T狸m gi叩 tr畛 nh畛 nh畉t c畛a bi畛u C担 s担短 boi d旦担探ng va棚n ho湛a- luye辰n thi 単a誰i ho誰c Cao Nguye但n-128/39 YwangBMT Trang 2
- 3. C担 s担短 BDVH - LTH Cao Nguye但nBMT www.luyenthicaonguyen.com C: 128/39 Ywang - BMT T: 0984.959.465-0945.46.00.44 bi畛u th畛c P = x y z + + 2x + 3y y + z z + x 34 x = 4; y = 1; z = 2 33 Bi 22 (H B2011) Cho c叩c s畛 th畛c a, b, l c叩c s畛 th畛c d動董ng th畛a m達n i畛u ki畛n : 錚 a 3 b3 錚 錚 a 2 b 2 錚 2(a 2 + b 2 ) + ab = (a + b)(ab + 2) . T狸m gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c P = 4 錚 3 + 3 歎 9 錚 2 + 2 歎. a 錚 錚b a 錚 錚b 錚a = 2 錚a = 1 23 S : MinP = 錚 ho畉c 錚 4 錚b = 1 錚b = 2 S : MinP = Bi 23 (H D2011NC) T狸m gi叩 tr畛 l畛n nh畉t v nh畛 nh畉t c畛a hm s畛 tr棚n o畉n [ 0; 2] . y= 2 x 2 + 3x + 3 x +1 17 S : Miny = y (0) = 3 ; Maxy = y (2) = [ 0;2] 3 [ 0;2] Bi 24 (H A2012) Cho c叩c s畛 th畛c x, y, z th畛a m達n i畛u ki畛n x +y + z = 0. T狸m gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c P = 3 x y + 3 y z + 3 z x 6 x 2 + 6 y 2 + 6 z 2 . S : MinP = 3 x = y = z = 0 Bi 25 (H B2012) Cho c叩c s畛 th畛c x, y, z th畛a m達n c叩c i畛u ki畛n x + y + z = 0 v x 2 + y 2 + z 2 = 1. T狸m gi叩 tr畛 l畛n nh畉t c畛a bi畛u th畛c P = x 5 + y 5 + z 5 . 5 6 6 6 x= ;y=z= 36 3 6 Bi 26 (H D2012) Cho c叩c s畛 th畛c x, y th畛a m達n (x 4)2 + (y 4)2 + 2xy 32. T狸m gi叩 tr畛 nh畛 nh畉t c畛a bi畛u th畛c A = x3 + y3 + 3(xy 1)(x + y 2). 17 5 5 1+ 5 S : MinA = x= y= 4 4 Bi 27 (H A2013) Cho c叩c s畛 th畛c d動董ng a, b, c th畛a m達n i畛u ki畛n (a + c)(b + c) = 4c 2 . T狸m gi叩 tr畛 S : MaxP = nh畛 nh畉t c畛a bi畛u th畛c P = 32a 3 32b3 a 2 + b2 + (b + 3c)3 (a + 3c)3 c S : MinP = 1 2 x = y = 1 Bi 28 (H B2013) Cho a, b, c l c叩c s畛 th畛c d動董ng . T狸m gi叩 tr畛 l畛n nh畉t c畛a bi畛u th畛c : 4 9 P= a 2 + b 2 + c2 + 4 (a + b) (a + 2c)(b + 2c) 5 S : MaxP = a = b = c = 2 8 Bi 29 (H D2013) Cho x, y l c叩c s畛 th畛c d動董ng th畛a m達n i畛u ki畛n xy y 1 . T狸m gi叩 tr畛 l畛n nh畉t x+y x 2y c畛a bi畛u th畛c: P = 2 2 6(x + y) x xy + 3y 5 7 1 + x= ;y=2 3 30 2 Bi 30 (H D2013NC) T狸m gi叩 tr畛 l畛n nh畉t v gi叩 tr畛 nh畛 nh畉t c畛a hm s畛 tr棚n o畉n [ 0; 2] . S : MaxP = 2 x 2 3x + 3 x +1 S : Minf(x) = f (1) = 1 ; Maxf(x) = f (0) = 3 [ 0;2] [ 0;2] f ( x) = C担 s担短 boi d旦担探ng va棚n ho湛a- luye辰n thi 単a誰i ho誰c Cao Nguye但n-128/39 YwangBMT Trang 3