How do I solve this function?

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leijonah

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Oct 21, 2005
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2^n+5^n=3^n+4^n where n=real numbers

and

limit as x-> positive infinity of f(x)

f(x)= x- [(x^pie)+(5x^(pie-1))+1]^(pie/2)
 
2^n+5^n=3^n+4^n where n=real numbers
Click to expand...
n=0 and n=1 are your obvious ones by inspection.

Not sure about others, perhaps numerical methods are necessary?
 
I have the basis of the second part of the question being an infinity-infinity. Therefore you would use l'hospitals rule....for indeterminate types, but my problem is how? would I find common denominator, apply natural logs, rationalize? Any input?
Thanks for your help
 
Now for the first part I understand n=0, n=1....now if I use Mean Value theorum to prove it, where would I start?
 
I have the basis of the second part of the question being an infinity-infinity. Therefore you would use l'hospitals rule....for indeterminate types, but my problem is how? would I find common denominator, apply natural logs, rationalize?
Click to expand...

Intuitively, it is clearly going to negative infinity (look at how fast the right-hand part will increase compared to the left).

You could try multiplying both terms by \(\displaystyle \frac{(x^\pi+5x^{\pi-1})^{\frac{\pi}{2}}}{(x^\pi+5x^{\pi-1})^{\frac{\pi}{2}}}\) and using L'Hopitals' but it's messy and doesn't necessarily lead to a better result than multiplying by \(\displaystyle \frac{x}{x}\).
 
Intuitively, it is clearly going to negative infinity (look at how fast the right-hand part will increase compared to the left).

You could try multiplying both terms by \(\displaystyle \frac{(x^\pi+5x^{\pi-1})^{\frac{\pi}{2}}}{(x^\pi+5x^{\pi-1})^{\frac{\pi}{2}}}\) and using L'Hopitals' but it's messy and doesn't necessarily lead to a better result than multiplying by \(\displaystyle \frac{x}{x}\).
Click to expand...

If in doing this it should reduce and I will get my answer of negative infinity, but if I had to use l'hospitals, I would multiply both parts of the functions to get the same denominator?? I just need clarity in what would be the most effective way of dealing with this equation...I am stuck...it just seems more complicated..
Thanks again
 
thanks, you wouldnt have any idea of how to use the mean value theorum for the first one, I already had that n=0,1 just by looking at it. But i have to prove it, and I want to use mean value theorum, but I am not certain as to where to start?
 
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