# How do you use the second derivative test to find where the function #f(x) = (5e^x)/(5e^x + 6)# is concave up, concave down, and inflection points?

##### 1 Answer

*concave down* on *concave up* on

#### Explanation:

The **second derivative test** allows you to determine the intervals on which a function is *concave up* or *concave down* by examining the sign of the second derivative around the **inflexion points**.

Inflexion points are points for which

So, start by calculating the first derivative of

Next, calculate the second derivative by using the quotient and chain rules

FInd the *critical poin(s)* of the function by calculating

This will get you

Take the natual log of both sides of the equation to get

Now investigate the sign of the sign derivative for values *smaller* than *larger* than

SInce

So, the two intervals that you're going to look at are

#(-oo, ln(1.2))#

For values smaller than **negative**, which means that **positive**.

As a result, **concave up** on this interval.

#(ln(1.2), +oo)#

This time, **negative**.

This implies that **concave down** on this interval.

The point *Inflexion point* for this function.