Algegra 挑戰題

a^2+b^2+c^2=2
(a^2+b^2+c^2)^2=2^2
a^4+b^4+c^4=4=4-2(a^2b^2+a^2c^2+b^2c^2)
in order to find the sup of a^4+b^4+c^4
4-2(a^2b^2+a^2c^2+b^2c^2) must be largest
2(a^2b^2+a^2c^2+b^2c^2) must be smallest
thus 2(a^2b^2+a^2c^2+b^2c^2)=0 when a=b=0 or b=c=0 or a=c=0
therefore, sup a^4+b^4+c^4=4

let f(a,b,c)=a^4+b^4+c^4
let g(a,b,c)=a^2+b^2+c^2=2
grad f = (4a^3,4b^3,4c^3)
grad g = (2a,2b,2c)
by Lagrange multipliers
4a^3=2Xa -(1)
4b^3=2Xb -(2)
4c^3=2Xc -(3) where X is some constant
from 1,2,3
2a^2=X -(4)
2b^2=X -(5)
2c^2=X -(6) where a!=0,b!=0,c!=0 (for a=0/b=0/c=0 case it is the max case which shown in above)
(4)+(5)+(6):
2(a^2+b^2+c^2)=3X
X=4/3
sub X=4/3 into 4,5,6
a^2 = 2/3 ->a^4=4/9
b^2=2/3 ->b^4=4/9
c^2=2/3 ->c^4=4/9
f(a,b,c)=a^4+b^4+c^4 = 4/3
the inf of a^4+b^4+c^4 =4/3
the range of a^4+b^4+c^4 is [4/3,4]

For simple,
Let A=a², B=b², C=c²
i.e. A+B+C=2

as (A+B+C) ²=A²+B²+C²+2(AB+BC+AC)
and 2(AB+BC+AC)≥0
∴(A+B+C) ²≥A²+B²+C²
i.e. A²+B²+C²≤2²=4
∴the maximum value of A²+B²+C²=4

Assume A≤B≤C
Let A=Z+α , B=Z+β and C=Z+γ
Where Z is a non-negative constant
And α ,β and γ are non-negative variables
∴A²+B²+C²
= (Z+α)²+( Z+β)²+( Z+γ)²
= 3Z²+2Z(α+β+γ)+α²+β²+γ²
∴when 2Z(α+β+γ)+α²+β²+γ² is minimum, then A²+B²+C² is also minimum
As 2Z(α+β+γ)+α²+β²+γ²≥0
when α=β=γ=0, 2Z(α+β+γ)+α²+β²+γ² is minimum and equals to zero
when α=β=γ=0, A=B=C=2/3
∴ when A=B=C=2/3, A²+B²+C² is minimum
i.e. the minimum value of A²+B²+C² = 3(2/3)² = 4/3

overall, A²+B²+C²∈ [4/3 , 4]
i.e. a⁴+ b⁴+ c⁴∈ [4/3 , 4]


i don't know whether my answer is correct
if there are any mistakes, can you help me to correct them, thank

it's such an interesting question !!!

(i) for any natural number n,
a^n larger than a if a is larger than 1
a^n smaller than a if a is between 1 and 0
(ii) for any a, no matter positive or negative,
a^2 and a^4 are positive.
(iii) by Cauchy-Schwarz Inequality,
(a*A+b*B)(a*A+b*B) smaller than (a*a+b*b)(A*A+B*B)
in words, bigger grouped with bigger will become much more bigger.
a^4 + b^4 + c^4 is greatest at b = c = 0,
in this way, a is most possibly large, and after duplicating itself, it becomes even larger.
Greatest value: ( sqrt(2) )^4 = 4
a^4 + b^4 + c^4 is smallest at a = b = c,
in this way, the value is distributed evenly among a,b,c and result in a value not large when compared to any other case.
Smallest value: 3* ( sqrt(2/3) )^4 = 4/3

Note that
(i) for any natural number n,
a^n larger than a if a is larger than 1
a^n smaller than a if a is between 1 and 0
(ii) for any a, no matter positive or negative,
a^2 and a^4 are positive.
(iii) by Cauchy-Schwarz Inequality,
(a*A+b*B)(a*A+b*B) smaller than (a*a+b*b)(A*A+B*B)
in words, bigger grouped with bigger will become much more bigger.


So, with the constraint a^2 + b^2 + c^2 = 2,

a^4 + b^4 + c^4 is greatest at b = c = 0,
in this way, a is most possibly large, and after duplicating itself, it becomes even larger.
Greatest value: ( sqrt(2) )^4 = 4

a^4 + b^4 + c^4 is smallest at a = b = c,
in this way, the value is distributed evenly among a,b,c and result in a value not large when compared to any other case.
Smallest value: 3* ( sqrt(2/3) )^4 = 4/3

2008-07-05 20:40:29 補充：
Brilliant!
my answer is kind of wild guess, and waiwaikeith's answer is apparently more elegant^^
nice work~