Concavity and Intervals from graphs.

[john3j]

Can someone help me understand where the concavity on this graph?


I think that f(x) concaves up on -5, concaves down on -5, and concaves upward on 5. How do I tell where the points of inflection are? Would they be (-5, 2), (-2, -1), and (5, 3)?

Any help understanding this would be greatly appreciated!

Thanks,
John

 

[MarkFL]

I am assuming you are to write the intervals on which the concavity is up and where it is down. You have the correct points of inflection. These are the points on the curve where concavity changes, i.e., where the second derivative changes sign.

 

[john3j]

I am assuming you are to write the intervals on which the concavity is up and where it is down. You have the correct points of inflection. These are the points on the curve where concavity changes, i.e., where the second derivative changes sign.

So in order to write the intervals where concavity is up or down, do I just find the point on the x-axis where concavity changes?

Thanks,
John

 

[MarkFL]

Those points on the x-axis will be the end-points of your intervals, not counting \(\displaystyle \pm\infty\) on the far ends.

 

[srmichael]

Hi john3j.

It's critical that you understand how to look at a graph and decipher the following:

1) relative max/min (specific points)
2) absolute max/min (specific points)
3) where the graph is increasing/decreasing (range of x values)
4) concave up/down (range of x values)
5) points of inflection (specific points)

I assume this is an AP Calculus class you are taking. The AP exam makers LOVE asking this stuff. But they also throw in a wrinkle and usually do not provide the f(x) function but instead they usually give you the f'(x) function, that is, the graph of the derivative. Then they expect you to be able to know how to do the following 5 things I mentioned above. Keep in mind there are graphs thay may not even have some of these items, i.e. there may be no extrema, or it may be increasing or decreasing for all x, or there may be no points of inflection.

Specifically to this problem of yours, like MarkFL correctly told you, the points of inflections are the points on a graph where the concavity changes. I usually tell the kids that I tutor to think of the point of inflection as a point on the graph where a "cap becomes a cup" or a "cup becomes a cap", that is, where the graph goes from concave down (cap) to concave up (cup) or vice versa.

Good luck!