# Precalculus : Derivatives

## Example Questions

### Example Question #10 : Determine Points Of Inflection

Find the point of inflection of the function .

Explanation:

To find the x-coordinate of the point of inflection, we set the second derivative of the function equal to zero.

.

To find the y-coordinate of the point, we plug the x-coordinate back into the original function.

The point is then .

### Example Question #11 : Determine Points Of Inflection

Determine the point(s) of inflection of .

and

No points of inflection exist.

No points of inflection exist.

Explanation:

The points of inflection exist where the second derivative is zero.

which can never be . Therefore, there are no points of inflection.

### Example Question #12 : Determine Points Of Inflection

Find the point(s) of inflection of .

No points of inflection exist.

No points of inflection exist.

Explanation:

The points of inflection will exist where the second derivative is zero.

.

This will never be , so there are no points of inflection.

### Example Question #13 : Determine Points Of Inflection

List the interval(s) where the function  is concave up.

The graph is always concave up.

The graph is never concave up.

Explanation:

The graph is concave up where the second derivative is positive. Let us first find out if there are any points of inflection to narrow our search.

.

Now we can perform the second derivative concavity test on points on either side of . Let us try .

The second derivative at  gives us  which is less than zero, so the graph is concave down in this interval. The second derivative at  gives us , which is positive. Hence, the graph is concave up on the interval .

### Example Question #14 : Determine Points Of Inflection

Find the x-coordinate(s) of the point(s) of inflection of .

There are no points of inflection.

There are no points of inflection.

Explanation:

The points of inflection will only exist where the second derivative is zero.

Now  therefore, there are no points of inflection.

### Example Question #15 : Determine Points Of Inflection

Find the inflection point(s) of .

There are no points of inflection.

Explanation:

The points of inflection, if any exist, will be found where the second derivative is zero.

.

To find the y-coordinates, we simply plug the x-coordinates in to the original function.

.

So  is an inflection point.

Also, , so  is another inflection point.

### Example Question #16 : Determine Points Of Inflection

List the intervals and determine where the graph of  is concave up and concave down.

The graph is always concave down.

The graph is concave up on .

The graph is concave down on  and .

The graph is concave up on  and .

The graph is concave down on .

The graph is concave up on .

The graph is concave down on .

The graph is always concave up.

The graph is concave up on  and .

The graph is concave down on .

Explanation:

We need to set the second derivative equal to zero to determine where the inflection points are.

are the x-coordinates of our inflection points. Thus the intervals of concavity are , and . We can use  as our test points.

, so the graph is concave up on .

, so the graph is concave down on .

, so the graph is concave up on .

### Example Question #17 : Determine Points Of Inflection

Determine the points of inflection, if any, of the following function:

Explanation:

The points of inflection of a function are those at which its concavity changes. The concavity of a function is described by its second derivative, and when the second derivative is 0 a point of inflection occurs. We find the second derivative of the function and then set it equal to 0 to solve for the inflection points:

So the function has only one point of inflection at x=5/3.

### Example Question #18 : Determine Points Of Inflection

Which of the following is an -coordinate of an inflection point of the graph of the following function?

Explanation:

The inflection points of a function are the points where the concavity changes, either from opening upwards to opening downwards or vice versa.  The inflection points occur at the x-values where the second derivative is either zero or undefined.  That means we need to find our second derivative.

We start by using the Power Rule to find the first derivative.

Taking the derivative once more gives the second derivative.

We then set this derivative equal to zero and solve.

This factors nicely.

Therefore our second derivative is zero when

8 is the only one of these two amongst our choices and is therefore our answer.

### Example Question #19 : Determine Points Of Inflection

Determine the points of inflection for the following function.

None