Concavity and Inflection Points II

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In summary, the conversation discusses a problem set and requests help with a specific question. The first derivative is calculated incorrectly, leading to incorrect results for finding inflection points and asymptotes. The correct first derivative is provided and the simplification is left to the listener.
  • #1
ardentmed
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Hey guys,

I'm having trouble with this problem set I'm working on at the moment. I'd appreciate some help with this question.

I'm only asking about two. Please ignore question one.
08b1167bae0c33982682_22.jpg


f'(x) = 2(x^ 1/3) - 1.

Moreover, I proceeded to find f'(x)=0 and f'(x) = DNE, which gave me f(2^1/3)=4.74 (I highly doubt that this is right).

As for inflection points, since f''(x) = (-2/3)x^(-4/3), no inflection points exist since x is only in the denominator.
For asymptotes, I took lim x-> 0 for the vertical asymptote and got x=0. . Therefore, a vertical asymptote exists at x=0, and lim x-> infinity gave me infinity, so there is no horizontal asymptote.

Am I close?

Thanks in advance for all the help guys.

Cheers,
ArdentMed.
 
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  • #2
Your first derivative is wrong, the critical number is actually a whole number :)
$$f(x) = 3x^{\frac{2}{3}} - x$$

Applying the power rule:
$$f'(x) = 3(\frac{2}{3})x^{\frac{-1}{3}} - 1$$

I'll leave the simplifying for you to do.
 

FAQ: Concavity and Inflection Points II

What is concavity and how is it related to inflection points?

Concavity is a measure of the curvature of a function. It describes the shape of the graph of a function and can be either positive (curving upward) or negative (curving downward). Inflection points are points on a function where the concavity changes, from positive to negative or vice versa.

How do you determine the concavity of a function?

The concavity of a function can be determined by taking the second derivative of the function. If the second derivative is positive, the function is concave up, and if it is negative, the function is concave down.

What is the significance of inflection points?

Inflection points are important because they mark the points where the concavity of a function changes. They can also be used to identify the maximum and minimum points of a function.

Can a function have more than one inflection point?

Yes, a function can have multiple inflection points. This occurs when the concavity of the function changes more than once.

How do inflection points affect the behavior of a function?

Inflection points can have a significant impact on the behavior of a function. They can indicate where the function is increasing or decreasing at an increasing or decreasing rate, and they can also help identify the points of maximum and minimum values of a function.

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