MHB Can the Inequality Challenge be Proven: 2^{\frac{1}{3}}+2^{\frac{2}{3}}<3?

AI Thread Summary
The discussion centers on proving the inequality \(2^{\frac{1}{3}} + 2^{\frac{2}{3}} < 3\). The proof begins by establishing that \(2^{\frac{1}{3}} > \frac{5}{4}\), leading to the conclusion that \(\frac{1}{2^{\frac{1}{3}} - 1} < 4\). This allows for the derivation that \(2^{2/3} + 2^{1/3} + 1 < 4\), which implies \(2^{2/3} + 2^{1/3} < 3\). However, a participant challenges the completeness of the proof, suggesting further analysis of the function's roots is necessary. The conversation emphasizes the importance of rigorous mathematical proof in addressing inequalities.
anemone
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Prove $$2^{\frac{1}{3}}+2^{\frac{2}{3}}<3$$.
 
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[sp]$2 = 128/64 > 125/64$, so (taking cube roots) $2^{1/3} > 5/4$ and $2^{1/3} -1 >1/4$. Therefore $\dfrac1{2^{1/3} -1} < 4$. But $$1 = 2-1 = (2^{1/3})^3 - 1 = (2^{1/3} -1)(2^{2/3} + 2^{1/3} + 1),$$ and so $2^{2/3} + 2^{1/3} + 1 = \dfrac1{2^{1/3} -1} < 4$. Thus $2^{2/3} + 2^{1/3} < 3$.[/sp]
 
Opalg said:
[sp]$2 = 128/64 > 125/64$, so (taking cube roots) $2^{1/3} > 5/4$ and $2^{1/3} -1 >1/4$. Therefore $\dfrac1{2^{1/3} -1} < 4$. But $$1 = 2-1 = (2^{1/3})^3 - 1 = (2^{1/3} -1)(2^{2/3} + 2^{1/3} + 1),$$ and so $2^{2/3} + 2^{1/3} + 1 = \dfrac1{2^{1/3} -1} < 4$. Thus $2^{2/3} + 2^{1/3} < 3$.[/sp]

Thanks for participating, Opalg! I really admire your talent in approaching this type of problem using the way you did.

My solution:

Let $$y=2^{\frac{1}{3}}+2^{\frac{2}{3}}$$. We're then asked to proved that $y<3$.

Then $$y^3=2+3(2^{\frac{1}{3}})(2^{\frac{2}{3}})(2^{\frac{1}{3}}+2^{\frac{2}{3}})+2^2=6+6y$$

$$y^3-6y-6=0$$

If we let $f(y)=y^3-6y-6$, we see that $f(2)=-10$ and $f(3)=3$, hence by the Intermediate Value Theorem, $y$ must have a solution between 2 and 3, i.e. $y<3$ and so we're done.
 
anemone said:
Thanks for participating, Opalg! I really admire your talent in approaching this type of problem using the way you did.

My solution:

Let $$y=2^{\frac{1}{3}}+2^{\frac{2}{3}}$$. We're then asked to proved that $y<3$.

Then $$y^3=2+3(2^{\frac{1}{3}})(2^{\frac{2}{3}})(2^{\frac{1}{3}}+2^{\frac{2}{3}})+2^2=6+6y$$

$$y^3-6y-6=0$$

If we let $f(y)=y^3-6y-6$, we see that $f(2)=-10$ and $f(3)=3$, hence by the Intermediate Value Theorem, $y$ must have a solution between 2 and 3, i.e. $y<3$ and so we're done.

anemone,
the proof provided by you is far from complete
(because if we take
f(x) = (y-1.5)(y-2.5)(y+.5) = 0

we get f(2.0) < 0 and f(3) > 0 and it has 3 roots)

based on this we need to prove in your case
either it has no other real solution or other 2 solutions if real are no where near 2^(1/3) + 2^(2/3)
 
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