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Chuyên đề sử dụng đạo hàm để tìm GTLN, GTNN

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SỬ DỤNG ĐẠO HÀM ĐỂ TÌM GIÁ TRỊ LỚN NHẤT, NHỎ NHẤT
CỦA HÀM SỐ
ITÓM TẮT LÍ THUYẾT


  
  
 
    
    
≤ ∀ ∈


∃ ∈ =


  
 
  

=

  
  
 
    
    
≥ ∀ ∈


∃ ∈ =




  
 
  

=
IIPHƯƠNG PHÁP GIẢI TOÁN
 !""#$%&"'
()*+,-."/'01+23'01+
345(
 / 
  
 
 
6
 / 
  
 
 
7"8&
9:$'."';-."/'8%<60-
+:-."/'0=<%(>+<&#<1"
<&0+"+?>"<10+"@>;5
-."/'
 ()*+,&A/'B6C0=D&
1?&&045
A / B
  
 

 
6
A / B
  
 
 

E1".'(01$%,"".;'F
++:$'."';&A/'B8%<60-+:
 !
"#$%&

'

'('

)*"+,&-./
"0*"'*

"'*

"'('*

"'*"
1"
( )
( ) ( )
( )
{ }
G H

A / B
      
 
         =
.
( )
( ) ( )
( )
{ }
G H
A / B
      
 
         =
IIIBÀI T)P ÁP DỤNG
B*i 1
I9I;J:$K"LLM"-

K H
K N KO   = − − +
&A2P/PB
2
0
 H
K L N  = − −

 H
G
Q K L N Q
K


  

= −

= ⇔ − − = ⇔

=

."(R=&A2P/PB
*2P)2PG*2G)PQ*K)S*P)GO
T:(
A P/PB
   PQ 

=
6
A P/PB
   PG 

= −

B*i 2
I9I;J:$K%"LLM"-

 H  = −
&
/
H H
π π

 

 
 
2
0

HH G = −


H H
G
L
Q HH G Q  H
H
H H
L
 3
  
 2
π
π
π
π

= +

= ⇔ − = ⇔ = ⇔



= − +


&
/
H H
π π
 

 
 
$ U"0"V+H)W
K
π
L

π
⇔ = ±
K K
  
H H H H L H L L H L
   
π π π π π π π π
       
− = = − − = − + = −
 ÷  ÷
 ÷  ÷
   
   
T:(()

H
π
()
H
π
B*i 3"+?>"@>;J:$K'"LLM"-

H
K H  = − +
&A2GQ/GQB
J+
0

[ ] [ ]
[ ]
H
H
K H GQ/G H/GQ
K H G/H
  

  

− + ∈ − ∪

=

− + − ∈




[ ] [ ]
[ ]

H K GQ/G H/GQ
H K G/H
 

 

− ∈ − ∪

=

− + ∈


J."'

2GQG
K
H
HGQ


2Q 2 
(
GKH
G
P

XH
 
QQ
6'."'(R";(+AQ/GKHB

A / B
   GKH
 
  =

A / B
   Q
 
  =
B*i 4
"+?>"@>;

( )
G H G H   = + + +
2
0
( )
H
L P   H G H   H H      = + + + + + +
E#) 
H ≤

H
G  H ⇒ = +
Y0

( )
H H
  H H H G H K      = = + − + +
6?
H ≤
0

H
HH P H + +
6?
G K G K
H/ / H
H H

   
− − − +
∈ − ∪
   
   
")

H
P S− +
6?
G K G K
/
H H

 
− − − +


 
 


G K G K
S G H/ / H
H H
 
G K G K
S  /
H H
 
 
 

   
− − − +
+ ∈ − ∪

   
    
=

 
− − − +

− ∈
 


 

J."'


H−

G K
H
− −
Q
G K
H
− +

H
( )

 
2Q2 
"


P H K−
6'."'(R";"+
H
P H K/P H G
 
− +
 

*ZQ
( )
*  ) P H K ) K G− −
/
( )
H
*  )P H G+
B*i 5
I[
G

H
+"V;$ U"

H H
H
GH
GH L P Q  

− + − + =

K K
G H
 +
E&I
'E&I9
2!
E1$ U"0"V
 H
Q P GH H K/ H H/H K 

   
⇔ ∆ ≥ ⇔ ≤ ≤ ⇔ ∈ − − ∪
   
\+]T2\
G H
H

 + =

H
G H
H
G GH
 P
GH
  

 
= − +
 ÷
 
0
( )
K K K
G H G H G H G H
K
  K 
H H

4        


= + = + − + = −
H
P H G
+
P H K+
S
H
P H G−


H
G K
Q Q
H H
4 

= + > ∀ ≠

J."'

23
H K−
2HQH
H K
3

4
 
^


G
P

0^+?>)
K K
P
^@>)
K K
P

B*i 6
I96I;

L L
P P
G  
G  
 

 
+ +
=
+ +
2
0
H
H
K
H  H

P
G
H  H
H




=

E#
[ ]
H
 H  Q/G5  5= ∈
0
( )
K
H
S K K S
P
G
S H H S
H
H
5
5 5
 4 5
5 5
5


− −
= = = =
− −

M(
( )
( )
[ ]

H
S
Q Q/G
H S
4 5 5
5

= < ∀ ∈

^_"'&AQ/GB0
I9;^_)^Q)GI9;^_)^G)
O
L
B*i 7
$ U"
H H
H H G P K Q    + + + + + =
I[
G

H

+"VI9;
G H G H
H 6    = − +
2!
E1$ U"0"V
H
Q L O Q O G  ⇔ ∆ ≥ ⇔ + + ≤ ⇔ − ≤ ≤ −
\+]T`\0
( )
H
G H G H
P K
G  
H
 
    
+ +
+ = − + =
K K
P

G
P
K K
P
( )
( )
H
H H
P K G G

H G S X S X
H H H
 
6     
+ +
= + + = + + = − + +
&A2O/2GB
0

P6 = − −
&A2O/2GBJ."'
 2O2P2G

6
Q2
a

N
H

PQ
T:(I9;a'b"
N
H
789#:;<':<<)=>
B*i 8 

H H
G + =
;

H
H
H 
H H G
 
?
 
+
=
+ +
2
H H H
H H H H H H
H  H  H 
H H G H H   K H
     
?
        
+ + +
= = =
+ + + + + + +
()QM)Q
(cQ.d6e;M
H

0
( )
H
H
H G

H G
K H G
K H G



?
 
 
 
 
+
 ÷
+
 
= =
+ +
 
+ +
 ÷
 
6?



=
0
( )
H


H H
H K L G
K H G
 
?
 
− + +
=
+ +


G H
K L K L
Q 
K K
?  
− − − +
= ⇔ = =
f#-
( )
H
H G
+ Q
K H G


 
→∞
+
=

+ +
J."'
 23
G

H
3

?
2QQ2
M
Q
H L
H
+

H L
H


Q
6'."'(
H L

H
?

=

H L


H
?
+
=
B*i 9
"@>;
P P H H
P P H H
     
4
     
 
= + − + + +
 ÷
 
T?'cQ
2
E#
 
5
 
= +

H  Q5  ≥ ∀ ≠
Y0
( ) ( )
H
H H P H
H H H O P4 5 5 5 5 5 5= − − − − + = − + +

 K
P GQ G4 5 = − +

 H
GH GQ4 5= −
M(
_ZH
( ) ( )
  
Q H GK Q4 4 5 4⇒ > ⇒ ≥ = >
_g2H
( ) ( )
  
Q H GG4 4 5 4⇒ > ⇒ ≥ − = −
J."'

_ 232HH3

4
2 
^
33
2HH
I@>;^)2H
B*i 10
(ZQ6()GI9I;

G G
 
?

 
= +
+ +
2
h".(
G
Q
P
≤ ≤
0
H H
G
   
?
  
+ + +
=
+ + +

( )
H
H G
G
  
  
+ − +
=
+ + +

H H

H



=
+
E#()
G
Q
P
≤ ≤
(
H H
H

?


=
+
( )

H
L
Q H
H
? 


= ≤ ∀ ≠ −

+
J."'

6'."'(I9;M'b"G/I;M'b"
H
K
B*i 11
ZQ
<(55(ZG
()K
"@>+?>;'1Fi)
K
H(
H
K
H
P(2O
2
ZQ
()K)j()K20)jQggH
(ZG
i)
K

H
`OH(
H
H(
H
)

K

H
`OH(
H
)
K

H
`OGS
_k*)
K

H
`OGS/
AQ/HB∀ ∈
*l)K
H
H`O/*l)Qm)j
G
O
K


=



= −


/
O
K
 = −
+&
0*Q)GS/*G)GO/*H)HQ
T:(I9;i'b"HQ/I;i'b"GO
J:$<".
JGT?";

H
O P   = − + +
0I+?UGn
JH$o1"+?>;

H
  9 = + +

&A2G/GB+'k>
JK(jQ6()G
I;
G G
 
?
 
= +
− −
JPI.d
H
H

G
Q 9
9
+ + =
0"V
G

H
$cQM)
G
P

H
P
@>

×