The intervals where a function is concave up or down is found by taking second derivative of the function. Use the power rule which states:

Now, set  equal to  to find the point(s) of infleciton.

In this case, .

To find the concave up region, find where  is positive. This will either be to the  left of   or to the right of . To find out which, plug in a test point in each of those regions.

If we plug in  we get , which is negative, so that cannot be concave up.

If we plug in , we get , which is positive, so we know that the region  will be concave up.

To find which interval is concave down, find the second derivative of the function.

Now, find which  values in the interval specified make . In this case,  and .

Now to find which interval is concave down choose any value in each of the regions

, and

and plug in those values into  to see which will give a negative answer, meaning concave down, or a positive answer, meaning concave up.

A test value of  gives us a  of . This value falls in the range, meaning that interval is concave down.

The derivative of  is

The derivative of this is

This is the second derivative.

A function is concave down if its second derivative is less than 0.

 whenever 

This is true when:

To find the concavity of a graph, the double derivative of the graph equation has to be taken. To take the derivative of this equation, we must use the power rule,  

.  

We also must remember that the derivative of an constant is 0.

After taking the first derivative of the equation using the power rule, we obtain 

. 

The double derivative of the equation we are given comes out to 

.  

Setting the equation equal to zero, we find that .  This point is our inflection point, where the graph changes concavity.  In order to find what concavity it is changing from and to, you plug in numbers on either side of the inflection point. if the result is negative, the graph is concave down and if it is positive the graph is concave up.  Plugging in 2 and 3 into the second derivative equation, we find that the graph is concave up from  and concave down from .

To find the invervals where a function is concave down, you must find the intervals on which the second derivative of the function is negative. To find the intervals, first find the points at which the second derivative is equal to zero. The first derivative of the function is equal to 

.

The second derivative of the function is equal to 

.

Both derivatives were found using the power rule 

. 

Solving for x, . The intervals, therefore, that we analyze are  and .

On the first interval, the second derivative is negative, which means the function is concave down. On the second interval, the second derivative is positive, which means the function is concave up. (Plug in values on the intervals into the second derivative and see if they are positive or negative.)

Thus, the first interval  is the answer. 

Points of inflection occur where there second derivative of a function are equal to zero. Taking the first and second derivative of the function, we find:

To find the points of inflection, we find the values of x that satisfy the condition

.

Which occurs at 

Within the defined interval [-5, 5], there are three values: .  These points are represented on the figure below as red dots.

To find the concavity of a function, we must take the second derivative of the function and set it equal to zero. 

Solving for , the point of inflection occurs at :

We must find where  is negative, which will either be to the left or the right of . Test points can be used to determine this.

Because the region less than  is negative, we know that the region from  is concave down.

Is the following function concave up or concave down on the interval [2,4]?

A function is concave up if its second derivative is positive, and vice-versa. so, we want to find h''(t).

Next, plug in the endpoints of our interval to find the sign of h''(t).

Since our second derivative is negative on the interval, we can say that h(t) is concave down on the interval.

In order to solve this we must visualize a triangle that has formed between the tip of the radio tower and the plane.  To find the distance the plane is from the tip of the tower, we must use the Pythagorean theorem  where  is the horizontal distance from the tower, in this case 300; and  is the vertical distance from the tip of the tower, in this case 400.  Therefore the distance the plane is away from the tip of the tower is 500 feet.  Now in order to find the horizontal speed of the airplane, we must take the derivative of the Pythagorean theorem with respect to time in order to find the change in horizontal distance of the plane with respect to time.  

Using the power rule 

,

we find that the theorem becomes 

.  

We need to find  and we know the variables  we simply plug in to find the answer.

This parabola would have the formula . When the first derivative is positive, the function is parabola is increasing. The first derivative is , which is positive on the domain . When the second derivative is positive, the function is concave up. The second derivative is , which is always positive for all real values of .  

Therefore, this function is,

Concave up over  and increasing over .