# How do you determine whether the function #f(x)= 2x^3-3x^2-36x-7# 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 intervals on which a function is *concave up* or *concave down* by examining the sign of the second derivative around the **inflexion point(s)**.

Inflexion point(s) are determined by making the second derivative equal to **zero**.

If the second derivative is *positive* on a given interval, then the function will be **concave up** on the same interval. Likewise, if the second derivative is *negative* on a given interval, the function will be **concave down** on said interval.

So, calculate the first derivative first - use the power rule

Next, calculate the second derivative. Once again, use the power rule

Find the inflexion point by solving

Now look on how the second derivative behaves for values of *smaller* than *larger* than

The two intervals you're going to use are

#(-oo,1/2)#

For this interval, **negative**, which means that **concave down**.

#(1/2, +oo)#

For this interval, **positive**, which implies that **concave up**.

So, *concave down* on *concave up* on