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

#### shamieh

##### Active member
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.