Question

See the attachment

Answer 1

Do you have the answer? Is it 4/153 ?

Answer 2

No wait, 4/51 ?

Answer 3

I don't have the answer.

Probably your solution would work

Answer 4

I just used symmetry ,
a=b=c

Answer 5

Sure that would work ?

Answer 6

not sure! :|

Answer 7

@RnR

Answer 8

Would that be required to prove

Answer 9

I dont know. :/

Answer 10

http://www.wolframalpha.com/input/?i=minimize+%28%28+a%2F%28b%5E3+%2B54%29%29+%2B+%28+b%2F%28c%5E3+%2B54%29%29+%2B+%28+c%2F%28a%5E3+%2B54%29%29+%29+%2C+a%2Bb%2Bc+%3D+9%2F2

Answer 11

I was saying 0.784314 approx i.e. 4/51 , but guess the ans is 0.7082808

Answer 12

So my logic must be flawed ? @RnR

Answer 13

MAY BE

Answer 14

oh wait!
my answer is correct !

Answer 15

it says all non negative numbers.

Answer 16

But, remember we will have to prove and justify each and every step.

Answer 17

hmm..I don't know the precise reasoning, but in most questions like these, symmetry works i.e. a=b=c.
As I said, I don;t know the precise reasoning .

Answer 18

Well I accept what you said about symmetry; but without reasons that wouldn't work.

Answer 19

@ganeshie8

Answer 20

@mukushla can surely guide you, but he;s not online..
hmm
@experimentX also!

Answer 21

well you may want to wait till the experts tune in..

Answer 22

this is not easy ... if it has numerical values, then Lagrange multiplies should work.

Answer 23

@DarthTony
How can you say that
a=b=c=x
3x=9/2, x=1.5 minimum = 3/2

First prove that this statement is true.

Answer 24

I think i saw similar Q on M.SE ... let me google it.

Answer 25

Whats that?

Answer 26

@experimentX whats M.SE

Answer 27

http://math.stackexchange.com

Answer 28

Ohhhhhhhhhh

Answer 29

I don't think you can just solve it by AM-GM type or Cauchy Swartz type

Answer 30

finding the minimum value and proving it are two different cases.

Answer 31

there's similar question here
http://math.stackexchange.com/questions/191431/inequality-fracaa23-fracbb23-leq-frac12?rq=1

Answer 32

@dan815 I accept what you say in pts. 1 and 2 but still I don't find it sufficient to say that a=b=c will satisfy our answer. This is a difficult question and could not be solved with that ease.

Answer 33

here is a similar question ... this says to look for Cauchy-Swarz

Answer 34

can it be possible that a=b=0 and c=9/2.so that we get the minimum value??o is a non negative number

Answer 35

looks like this Q wen't unsolved in MSE too
http://math.stackexchange.com/questions/275208/the-least-value-for-fracab354-fracbc354-fracca354

Answer 36

what class is this from?

Answer 37

@Peter14
No class

Answer 38

looks like Olympiad style Q.

Answer 39

if it's just min value then try making use of Lagrange Multiplies. @DarthTony the criteria is slight different a+b+c = 9/2 .... so this might be equal your answer. If it's just finding the value without caring method, then Lagrange multiplies also work.

Answer 40

@experimentX
Correct said. Its an olympiad question

Answer 41

\[
f(a,b,c)=\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}\\
g(a,b,c)=a+b+c-{9\over2}\\

f_a=\lambda g_a=\lambda\\
f_b=\lambda g_b=\lambda\\
f_c=\lambda g_c=\lambda\\
a+b+c={9\over2}
\]
this is the set of langrangian equations
the first three derivatives must be fairly simple (each giving a relation between two variables and \(\lambda\)).

Answer 42

like this: the first equ.\[{1\over b^3+54}-{3a^2c\over (a^3+4)^2}=\lambda\]

Answer 43

hmm.. no global minima!!
lets see..
are the partial derivatives positive?

Answer 44

sorry again ... i was putting 24 against 54
http://www.wolframalpha.com/input/?i=Minimize%5B%7Ba%2F%28b%5E3+%2B+54%29+%2B+b%2F%28c%5E3+%2B+54%29+%2B+c%2F%28a%5E3+%2B+54%29%2C++++a+%2B+b+%2B+c+%3D%3D+9%2F2+%26%26+a+%3E%3D+0+%26%26+b+%3E%3D+0+%26%26+c+%3E%3D+0%7D%2C+%7Ba%2C+b%2C+c%7D%5D

Answer 45

if a function is bounded by numerical values, then Lagrange multiplies should work.

Answer 46

who is cheerful enough in the morning to continue the expansion? -> the real slim shady math guys say aye

Answer 47

i wonder if it can be proved by Cauchy Schwarz. If it's just asking for the value then Lagrange multiplies as well as guessing like symmetry argument.

@electrokid i give up on lagrange

Answer 48

gotta go for now.

Answer 49

@SheldonEinstein is this a numerical problem or an analytical one?

Answer 50

Can't say as it is an olympiad one.