To find intervals of concavity, we need to find the inflection points of f. To do this, we need to take the second derivative of  and find the values of  for which it is :

so....  

These are the two inflection points of . In order to find the intervals where  is concave down, we need to find the intervals where  is negative (this being true because a negative second derivative means the signed rate of change of the derivative is negative, which is true when we have a steepening negative rate of change or a leveling off positive rate of change for , consistent with an upside down bowl). Since  is continuous, we can find this out by taking test values in the three intervals created by the two inflection points.

Convenient values are ,  and ; one for each interval:

 = 

so  is concave down on the interval given in the answer. The reason there isn't a larger or different interval is:

    so concave up

and

 so concave up

In order to find the intervals of concavity, we must take the second derivative of the function  and find the inflection points by setting the setting it to zero (this tells allows us to see when the rate of the rate of change is changing from negative to positive or vice versa).  The second derivative is , allowing us to see that our inflection points are .  By testing values in between the three different intervals, we can find out which interval is concave down.  The interval  is concave down because placing any value in between these two numbers into  will provide a negative output.

Is the following function concave up or down on the interval ?

Recall that the test for cancavity is similar to the one for increasing/decreasing, but with a key difference. To test concavity, we must use the second derivative instead of the first.

Begin by putting f(x) in standard form and then find its derivative

Now that we have our second derivative, we need to try both our endpoints to see what kind of an answer we get:

So, both our endpoints yield negative numbers. If you think critically, you should be able to see that any number between 3 and 6 will also yield a negative number.

This means that f(x) is concave down on the given interval, because all possible values of its second derivative on the interval are negative.

To determine the intervals on which the function is concave down, we must find the intervals on which the second derivative of the function is negative. 

First, we must find the second derivative:

The derivatives were found using the following rule:

Now, we set the second derivative equal to zero and find the x values at which this is true:

Next, we make our intervals on which we determine if the second derivative is positive or negative, with 0 as the upper and lower bound for the intervals:

Note that at 0 the second derivative is neither positive nor negative.

On the first interval, the second derivative is always negative, while on the second interval, the second derivative is always positive (simply plug in any point on the interval into the second derivative and check the sign). Thus, on the first interval -  - the function is concave down.

To find the intervals with the same concavity, we need to find the critical points using the second derivative test, then see what the concavity is in the intervals using the second derivative.

; set equal to 0 and solve for , giving  as the only critical point.

Choose an -value either side of the critical point, and test the concavity. For example:

, so the graph is concave down to the left of the critical point.

, so the graph is concave up to the right of the critical point.

Therefore the function is concave down on the interval .

To solve, simply differentiate twice, using the power rule, and then find where the second derivative is negative.

Power rule:

Thus,

Since the second derivative is always negative, our function is concave down everywhere.

To find concavity, you must find the second derivative using the power rule:

Since we are looking for concave down, we are looking for when the second derivative is negative (less than 0). Thus,

Since 14 is always greater than 0, the second derivative is never negative. Thus, our function is never concave down.

To determine the intervals on which the function is concave down, we must find the intervals on which the second derivative of the function is negative.

First, we must find the second derivative:

The derivatives were found using the following rule:

Now, we must find the value at which the second derivative is equal to zero.

We will now use this as the upper and lower limit of our intervals on which we evaluate the sign of the second derivative:

On the first interval, the second derivative is negative, while on the second interval, the second derivative is positive. Thus, our answer is .

To find the intervals on which the function is concave down, we must find the intervals on which the second derivative of the function is negative.

First, we must find the first and second derivatives:

The derivatives were found using the following rule:

Next, we must find the values at which the second derivative is equal to zero:

Now, we can make the intervals:

Note that at the bounds of the intervals the second derivative is neither positive nor negative.

To determine the sign of the second derivative on the intervals, simply plug in any value on the interval into the second derivative function; on the first interval, the second derivative is positive, on the second it is negative, and on the third it is positive. Thus, the function is concave down on the interval .

Tell whether f(x) is concave up or concave down on the interval [1,2]

To find concave up and concave down, we need to find the second derivative of f(x).

Let's begin by finding f'(x)

Next find f ''(x)

Now, to test for concavity, plug in the endpoints of the interval:

So, on this interval, f"(x) will always be negative. This means that our function is concave down on this interval.