First, a basic inequality is proved: there are inequalities for any positive real number x, y, z y and z.
(x+y+z)( 1/x+ 1/y+ 1/z)>=9
Established.
Proof: (If you know Caychy inequality, then this is the direct inference of Cauchy inequality)
Multiply all expressions in two brackets, that is
(x+y+z)( 1/x+ 1/y+ 1/z)= 3+(x/y+y/x)+(x/z+z/x)+(y/z+z/y)。
The last three brackets use mean inequality, that is, x+y+y/x >; =2,
x/z+z/x & gt； =2,
y/z+z/y & gt； =2,
So we know
(x+y+z)( 1/x+ 1/y+ 1/z)>=3+2+2+2=9,
Therefore, the above basic inequality holds.
Back to the original question. According to the average inequality,
1+a^2=8/9+( 1/9+a^2)>； =8/9+2/3a，
In the same way; In a similar way
1+b^2>； =8/9+2/3b，
1+c^2>； =8/9+2/3c, so
a/( 1+a^2)
+b/( 1+b^2)
+c/( 1+c^2)
& lt= a/(8/9+2/3a)+b/(8/9+2/3b)+c/(8/9+2/3c)
= 3/2 *[3a/(3a+4)+3b/(3 b+4)+3c/(3c+4)]
( 1)
According to the above basic inequality:
[ 1/(3a+4)+ 1/(3 b+4)+ 1/(3c+4)]*[(3a+4)+(3 b+4)+(3c+4)]& gt； =9,
but
a+b+c= 1，
(3a+4)+(3b+4)+(3c+4)= 15，
therefore
1/(3a+4)+ 1/(3 b+4)+ 1/(3c+4)>=9/ 15=3/5
(2)
From (2)
4/(3a+4)+4/(3b+4)+4/(3c+4)>= 12/5,
Subtract both sides of the inequality from 3 to get:
3-[4/(3a+4)+4/(3 b+4)+4/(3c+4)]& lt； =3- 12/5=3/5,
that is
( 1-4/(3a+4))+( 1-4/(3 b+4))+( 1-4/(3c+4))& lt； =3/5,
therefore
3a/(3a+4)+3b/(3 b+4)+3c/(3c+4)& lt； =3/5
(3)
Substitute the result of (3) into (1).
a/( 1+a^2)
+b/( 1+b^2)
+c/( 1+c^2)
(Through (1))
& lt= 3/2 *[3a/(3a+4)+3b/(3 b+4)+3c/(3c+4)]
(Through (3))
& lt=(3/2)*(3/5)
=9/ 10
Therefore, the proved inequality holds.
Note: the key to the problem lies in the equal conditions. Note that the equal sign of the formula is proved to be true and only if.
a=b=c= 1/3，
So you need to use it in your proof.
a^2+ 1=a^2+ 1/9+8/9>； =2/3a+8/9。
Only in this way can the scoring conditions be consistent.
When one of a, b and c is 0, for example, c=0, the formula is
a/(a^2+ 1)+b/(b^2+ 1)<； =9/ 10,
The conditions that need to be met at this time are as follows
a+b= 1。
Then we can know from the above method that if and only if a=b= 1/2.
A/(A 2+ 1)+B/(B 2+ 1) takes the maximum value.
4/5 & lt； =9/ 10.