# How do you determine whether the function #f(x) = (x^2)/(x^2+1)# is concave up or concave down and its intervals?

##### 1 Answer

You can use the *second derivative test*.

#### Explanation:

The **second derivative test** allows you to determine the concavity of a function by analyzing the behavior of the function's second derivative around *inflexion points*, which are points at which

If *positive* on a given interval, then **concave up**. LIkewise, if *negative* on a given interval, then **concave down**.

The inflexion points will determine on which intervals the sign of the second derivative should be examined.

So, start by calculating the function's first derivative - use the quotient rule

Next, calculate the second derivative - use the quotient and chain rules

Next, find the *inflexion points* by calculating

This is equivalent to

Take the square root of both sides to get

So, the function's graph has two inflexion points at **three intervals** to determine its concavity.

Since

More specifically, if

#(-oo, -sqrt(3)/3)#

On this interval, **concave down**.

#(-sqrt(3)/3, sqrt(3)/3)#

On this interval, **concave up**.

#(sqrt(3)/3, +oo)#

Once again, **concave down**.

So, your function is *concave down* on *concave up* on

The function's graph will have *two inflexion points* at

graph{x^2/(x^2+1) [-4.932, 4.934, -2.465, 2.467]}