When is f concave up

However, unlike the previous section, this time as we draw an increasing or decreasing portion of the curve we will also pay attention to the concavity of the curve as we are doing this. At the same time, we know that we also have to be concave down in this range. We can use the previous example to illustrate another way to classify some of the critical points of a function as relative maximums or relative minimums. It is also important to note here that all of the critical points in this example were critical points in which the first derivative was zero and this is required for this to work.

Here is the test that can be used to classify some of the critical points of a function. The proof of this test is in the Proofs of Derivative Applications section of the Extras chapter. The third part of the second derivative test is important to notice. If the second derivative is zero then the critical point can be anything. So, we can see that we have to be careful if we fall into the third case. For those times when we do fall into this case we will have to resort to other methods of classifying the critical point.

This is usually done with the first derivative test. The second derivative is,. The value of the second derivative for each of these are,. Note however, that we do know from the First Derivative Test we used in the first example that in this case the critical point is not a relative extrema. This is a common mistake that many students make so be careful when using the Second Derivative Test.

Joshua Siktar. Determining concavity obviously requires finding the second derivative, if it even exists. Related Lessons. Rate of Increase of a Quadratic Function. Saddle Points and Turning Points.

View All Related Lessons. Alex Federspiel. So let's look at an example to see how this all works. If we plug in , we get , which is positive, so we know that the region will be concave up.

Find the intervals that are concave down in between the range of. Now, find which values in the interval specified make. In this case, and. A test value of gives us a of.

This value falls in the range , meaning that interval is concave down. The function is concave-down for what values of over the interval? The derivative of is. Given the equation of a graph is , find the intervals that this graph is concave down on. 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,. 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. 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. 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:. Which occurs at. Within the defined interval [-5, 5], there are three values:. These points are represented on the figure below as red dots. If you've found an issue with this question, please let us know.

With the help of the community we can continue to improve our educational resources. A point where the concavity changes from up to down or down to up is called a point of inflection POI ; note that the tangent line to a graph at a point of inflection must cross the graph at that point. Now let's look at concavity from a slightly different perspective. Consider the graph of a cost function shown below.

The graph shows that the total cost of a certain activity increases sharply at the beginning and then rises more and more slowly until a point when the total cost begins to rise more sharply again. The blue colour indicates a region where the slope of the tangent decreases.

That is, in this region the rate at which the cost function increases, decreases. The red colour indicates a region where the slope of the tangent increases, i.

By our previous definitions, the blue area is concave downward and the red area is concave upwards. The green point is the point at which the rate of change of the slope changes from decreasing to increasing. It is also the point at which the concavity of the function changes from downward to upward.