# Precalculus : Determine Points of Inflection

## Example Questions

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### Example Question #1 : Determine Points Of Inflection

Determine the location of the points of inflection for the following function:

Explanation:

The points of inflection of a function are the points at which its concavity changes. The concavity of a function is described by its second derivative, which will be equal to zero at the inflection points, so we'll start by finding the first derivative of the function:

Next we'll take the derivative one more time to get the second derivative of the original function:

Now that we have the second derivative of the function, we can set it equal to 0 and solve for the points of inflection:

### Example Question #2 : Determine Points Of Inflection

Find the points of inflection of the following function:

Explanation:

The points of inflection of a function are those at which its second derivative is equal to 0. First we find the second derivative of the function, then we set it equal to 0 and solve for the inflection points:

### Example Question #3 : Determine Points Of Inflection

Find the inflection points of the following function:

Explanation:

The points of inflection of a function are those at which its second derivative is equal to 0. First we find the second derivative of the function, then we set it equal to 0 and solve for the inflection points:

### Example Question #4 : Determine Points Of Inflection

Determine the points of inflection of the following function:

Explanation:

The points of inflection of a function are those at which its second derivative is equal to 0. First we find the second derivative of the function, then we set it equal to 0 and solve for the inflection points:

### Example Question #5 : Determine Points Of Inflection

Determine the x-coordinate of the inflection point of the function .

Explanation:

The point of inflection exists where the second derivative is zero.

, and we set this equal to zero.

### Example Question #6 : Determine Points Of Inflection

Find the x-coordinates of all points of inflection of the function .

There are no points of inflection

Explanation:

We set the second derivative of the function equal to zero to find the x-coordinates of any points of inflection.

, and the quadratic formula yields

.

### Example Question #7 : Determine Points Of Inflection

Determine the x-coordinate(s) of the point(s) of inflection of the function .

There are no points of inflection.

Explanation:

Any points of inflection that exist will be found where the second derivative is equal to zero.

.

Since , we can focus on . Thus

, and .

### Example Question #8 : Determine Points Of Inflection

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

There are no inflection points.

Explanation:

The inflection points, if they exist, will occur where the second derivative is zero.

### Example Question #9 : Determine Points Of Inflection

Find the point(s) of inflection of the function .

There is no point of inflection.

Explanation:

The point of inflection will exist where the second derivative equals zero.

.

Now we need the y-coordinate of the point.

Thus the inflection point is at .

### 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 .

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