Intervals of concavity and inflection points

$\displaystyle {x^{1/5}}-{x^{1/3}}$

I found the first and second derivatives. The second derivative was ugly so I checked the answer and it said to use a calculator to graph and get the zeros. The part I got stumped on was to make sure there is an inflection point at 0, where 2nd derv. is undefined, you have to take the limit of the first derv. approaching zero and see if there is a tagent line there. Not sure how to solve the limit.

$\displaystyle \lim_{x-0}\1/5{x^{4/5}-{1/{3x^{2/3}$

Re: Intervals of concavity and inflection points

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**wondering** $\displaystyle {x^{1/5}}-{x^{1/3}}$

I found the first and second derivatives. The second derivative was ugly so I checked the answer and it said to use a calculator to graph and get the zeros. The part I got stumped on was to make sure there is an inflection point at 0, where 2nd derv. is undefined, you have to take the limit of the first derv. approaching zero and see if there is a tagent line there. Not sure how to solve the limit.

$\displaystyle \lim_{x-0}\1/5{x^{4/5}-{1/{3x^{2/3}$

The fact that the second derivative does not exist at x= 0 tells you that limit will not exist. However, the definition of "inflection point" is that the first derivative changes sign there. You should be able to determine the sign for x close to 0 on both sides of 0.

Re: Intervals of concavity and inflection points

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**HallsofIvy** The fact that the second derivative does not exist at x= 0 tells you that limit will not exist. However, the definition of "inflection point" is that the first derivative changes sign there. You should be able to determine the sign for x close to 0 on both sides of 0.

I am self studying Calc. so I may not be up to speed on somethings. I dont understand how the 2nd derv. tells you about the limit. I understand what you mean about the sign change of the 1st derv. The book is telling me that the limit is infinity, which seems to make sense because I believe there was an asymptote there. I dont have the book with me at work so I am going off of memeory. If I had to solve the limit to show that it indeed was infinity how would I start?