International Journal of Trend in Scientific Research and Development (IJTSRD)
Special Issue on International Research Development and Scientific Excellence in Academic Life
Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470
Using Cauchy Inequality to Find Function Extremums
G. A. Akhmedova, O. Yu. Makhmudova
Kokand State Pedagogical Institute, Kokand, Uzbekistan
ABSTRACT
The article explains the relationships between the mean
values using triangles. Here are some ways to determine
the extreme values of some functions using the Cauchy
inequality, which represents the relationship between
arithmetic mean and geometric mean.
function
KEYWORDS:
inequality
!
#
ℎ
⇒
extremums,
averages,
=√
!
=
≥ √ . (The
It can be seen from Figure 2 that it is
arithmetic mean of and is equal to the mean geometry
at
= .) From the similarity of triangles
& and
!(
!
'
= ; ⇒ & = + or & =
; & <
because
!
≤√
!)
,
(fig. 3).
The three mean values are equal to
= .
Now let's talk about the average values application in
solving problems to find the largest values of the function.
1-example. At what value of / does the (/) = /(1 −
/) (0 < / < 1) function take the largest value?
Solution: According to
/(1 − /) ≤ 5
/(1 − /) =
7
6 786
:
9 =
⇔ /=
Answer: max (/) =
7
7
≥√
inequalities
7
7
:
2-example. At what value of / does the function
(/) = /(/ − 1) (0 < / < 1) take the smallest value?
Solution. If < < = ℎ
− < > −=.
/(/ − 1) = − /(1 − /) ≥ − 5
/(/ − 1) = −
7
:
⇔/=
Answer: min (/) =
7
6 786
5 9=−
7
. We can compare these values in
the following ways (Figure 1).
2-rasm
3-rasm
3-example. At what value of / does the function
B
ℎ(/) = 4/ + , / > 0 take the smallest value?
6
Solution. According to
4/ + ≥ 2D · 4/ = 12
B
B
6
6
4/ + = 12 ⇔ / =
B
6
+
≥ 2√
F
Answer: min ℎ(/) = ℎ 5 9 = 12
F
4-example. At what value of / does
function / > 0 take the largest value?
Solution:
:6
G
H
:6
= ⇔/=
7
I
G
H
F
≤D
7
:6
· =
6
B
Answer: max (/) = 5 9 =
:
5 9=
average harmonic is
Cauchy
1-rasm
and
Consider sections with lengths
and . Their average
arithmetic is
, their average geometry is √ , and their
7
:
9 =−
7
:
F
(/) =
+
G
H
, the
7
I
7
I
5-example. At what values of (/) =
function / > −2 takes the smallest value.
Solution: According to
:6
≥ 2√
6 , :6 JF
6
, the
/ + 4/ + 53
49
49
= /+2+
≥ 2N(/ + 2) ·
= 14
/+2
/+2
/+2
6 , :6 JF
6
= 14 ⇔ / = 5
Answer: min (/) = (5) = 14
6-example. Prove that a rectangle with equal sides has a
square with the smallest perimeter.
ID: IJTSRD38751 | Special Issue on International Research Development and Scientific Excellence in Academic LifePage 114
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
Let one side of the rectangle be the one length, then the
other side of the given surface O =
( > 0) is
,
6
.
Proof: Let one side of the rectangle be the one length, then
the other side of the given surface O =
( > 0) is
This is the rectangle perimeter
P(/) = 2(/ +
It’s clear,
2 5/ +
2 5/ +
,
6
6
)
9 ≥ 2 ∙ 2D/ ∙
,
6
,
6
.
7-example. If A =
Proof. A =
,
6
=4 .
9= 4 ⇔/ −2 /+
min P(/) = 4
,
6,
if = / = . In that case, the rectangles with equal sides
6
have the smallest perimeter.
x2 + 3
x2 + 3
x +2
2
x2 + 2
=
,
B = 2 , then A > B .
x2 + 2 +
1
x +2
2
> 2.
So A > B . Proved.
=0⇔/=
References
[1] Sivashinsky I.Kh.
Science,1967
[2]
Inequalities
in
tasks.
M:
Zaitsev V. V., Ryzhkov V. V., Skanavi M. I. Elementary
mathematics. M: Science,1976
ID: IJTSRD38751 | Special Issue on International Research Development and Scientific Excellence in Academic LifePage 115