(
1
)
.
o
m
i
t
(1). omit
(1).omit
(
2
)
.
(
a
2
−
b
2
)
(
x
2
a
2
−
y
2
b
2
)
=
(
x
2
+
y
2
)
−
(
a
2
y
2
b
2
+
b
2
x
2
a
2
)
≤
x
2
+
y
2
−
2
x
y
=
(
x
−
y
)
2
(2). (a^2-b^2)(\frac{x^2}{a^2} - \frac{y^2}{b^2})=(x^2+y^2)-(\frac{a^2y^2}{b^2}+\frac{b^2x^2}{a^2}) \le x^2+y^2-2xy=(x-y)^2
(2).(a2−b2)(a2x2−b2y2)=(x2+y2)−(b2a2y2+a2b2x2)≤x2+y2−2xy=(x−y)2
- when a y b = b x a \frac{ay}{b}=\frac{bx}{a} bay=abx, the equation is satisfied.
(
3
)
.
(3).
(3).
s
e
t
:
3
m
−
5
=
x
,
m
−
2
=
y
set: \sqrt{3m-5}=x,\sqrt{m-2}=y
set:3m−5=x,m−2=y
t
h
e
n
:
x
2
−
3
y
2
=
1
then: x^2-3y^2=1
then:x2−3y2=1
h
a
v
e
:
x
2
1
−
y
2
1
3
=
1
,
have:\frac{x^2}{1}-\frac{y^2}{\frac{1}{3}}=1,
have:1x2−31y2=1, according to the question(2), we get:
(
x
−
y
)
2
≥
(
1
−
1
3
)
(
x
2
1
−
y
2
1
3
)
=
2
3
(x-y)^2 \ge (1-\frac{1}{3})(\frac{x^2}{1}-\frac{y^2}{\frac{1}{3}})=\frac{2}{3}
(x−y)2≥(1−31)(1x2−31y2)=32
- Pay attention to equal conditions!!!
- we can click here to see some more examples about the Cauchy-inequality.
