Question

for \(a,b,c \in \mathbb{R^{+}}\)
prove that:
\[(a^5-a^2+3)(b^5-b^2+3)(c^5-c^2+3)\ge \left (a+b+c \right)^3\]

Answer 1

\[(a^5-a^2+3)(b^3-b^2+3)(c^5-c^2+3) \ge (a+b+c )^3\]
just trying to retype it
having troubles looking at the other one

Answer 2

i just corrected it :)
should look fine now hahah

Answer 3

b^5 not b^3

Answer 4

\[f(x)=x^5-x^2+3 \\ f(x) >0 \text{ for } x>0 \\ \text{ and we have} \\ g(x)=x^3 \\ g(x) > 0 \text{ for } x>0 \\ \]
\[(f(x))^3=(x^5-x^2+3)^3 \\ =(x^2(x^3-1)+3)^3 \\ =(x^2(x^3-1))^3+3(x^2(x^3-1))^2(3)+3(x^2(x^3-1))(3)^2+3^3 \\ =x^5(x^3-1)^3+9x^4(x^3-1)^2+27x^2(x^3-1)+27 \]
I'm just thinking here ...

Answer 5

like I'm kinda just looking at the case a=b=c right now

Answer 6

\[g(x)=x^3 \\ g(3x)=(3x)^3=27x^3 \]

Answer 7

we should be able to show f^3>=g(3x) if this inequality does hold

Answer 8

hmm interesting!
i was trying AM-GM inequality
i was interested in this after i saw some questions related to that inequality

Answer 9

well I don't know if my way is good or not
and so far I haven't gotten anywhere quick yet

Answer 10

i didn't really go that far with am-gm either
but i think should work

Answer 11

anyways gotta go
have a paper to do lol

Answer 12

I have to think on this more later too
time to eat

Answer 13

Not sure if this is correct at all, but I graphed a representation of each side of the inequality to compare them, as can be seen in the attached screenshot. The expression on the left is the product of the same "function" evaluated three times at a, b, then c. So, I defined a function f(w) = w^5-w^2+3, chose values for constants a, b, and c, and then graphed the product f(x+a)f(x+b)f(x+c). I then graphed y=(x+a+b+c)^3. Clearly, the first equation (in blue in the screenshot) is greater than the second, if the graph is correct, that is. I tried picking different values for a, b, and c, and the first equation was still greater than the second. Again, I don't know if this approach is valid, but it is all I can think of at the moment.

Answer 14

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