Click here to see how to enable Java on your web browser. This applet is based on free Java applets from JavaMath. Given the above, we can decide whether a function is increasing or decreasing by looking at the sign of its derivative.
To decide whether f ' is increasing or decreasing, we must look at the sign of its derivative the derivative of f '. This derivative of f ' i. Examine the first example given below. Note that the function is shown on the left, the first derivative in the middle and the second derivative on the left. Examine the fourteen examples provided in the scroll bar on the top of the applet below or enter your own function in the box provided.
We can calculate the second derivative to determine the concavity of the function's curve at any point. Calculate f " x. 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. I'm not gonna do these out, but here's what you should get. 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.
We will need the list of possible inflection points. These are,. Here is the number line for the second derivative.
Note that we will need this to see if the two points above are in fact inflection points. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode.
If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Example 1 For the following function identify the intervals where the function is increasing and decreasing and the intervals where the function is concave up and concave down. Correct answer:. Explanation : To find when a function is concave, you must first take the 2nd derivative, then set it equal to 0, and then find between which zero values the function is negative.
First, find the 2nd derivative: Set equal to 0 and solve: Now test values on all sides of these to find when the function is negative, and therefore decreasing. Report an Error. Possible Answers: No, is positive on the interval. Correct answer: Yes, is negative on the interval.
So, for all So on the interval -5,-4 f x is concave down because f'' x is negative. Possible Answers: Concave down, because is negative on the given interval. Correct answer: Concave up, because is positive on the given interval. Explanation : To test concavity, we need to perform the second derivative test. Begin as follows: Next, we need to evaluate h" t on the interval [5,7] So, our h" t is positive on the interval, and therefore h t is concave up.
Explanation : 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.
Explanation : To find which interval is concave down, find the second derivative of the function. 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.
Possible Answers: It is never concave down. Explanation : 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. Explanation : To find the concavity of a graph, the double derivative of the graph equation has to be taken.
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. Determine the intervals on which the following function is concave down :.
Explanation : 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. The second derivative of the function is equal to. Both derivatives were found using the power rule. How many infelction points does the function have on the interval? Possible Answers: One.
Correct answer: Three. Explanation : 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.
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