Finding concavity when differentiating to dy/dx and d^2y/dx^2

[shamieh]

Suppose I have something like \(\displaystyle \frac{-2(t - 1)}{9t^5}\)

I know I just plug in two points to check its concavity...But How do I know what to choose between what and what? Like would I just choose between 0 and 1? And how do I know ?

Also suppose I have something like

\(\displaystyle - \frac{1}{4t^3}\)

would I pick a point that is to the right of and left -.25 so like 0 and -.5? and see that it always concaves up because it is cubed?

 

[MarkFL]

I am assuming the expressions you have given are the second derivatives of a given function. You want to look at the roots of the numerator and denominator as places where concavity may change. For $n$ total distinct roots, you will find the domain of the original function divided into $n+1$ sub-intervals.

So, you can pick a value from each of the resulting sub-intervals and test the second derivative to see whether it is positive (concave up) or negative (concave down).

Now, if you know a little about the nature of a function regarding its roots, you can use the fact that roots of odd multiplicity will be where sign changes occur while roots of even multiplicity will be where sign changes do not occur. This can result in your only needing to check one sub-interval and then inferring the sign of the second derivative in all of the other sub-intervals. This can save a lot of tedious work.