# Calculus 1 : Graphing Functions

## Example Questions

### Example Question #6 : How To Graph Functions Of Points Of Inflection

Find all the points of infection of

.

There are no points of inflection.

Explanation:

In order to find the points of inflection, we need to find  using the power rule .

Now lets factor .

Now to find the points of inflection, we need to set .

.

From this equation, we already know one of the point of inflection, .

To figure out the rest of the points of inflection we can use the quadratic equation.

Recall that the quadratic equation is

, where a,b,c refer to the coefficients of the equation

.

In this case, a=20, b=0, c=-18.

Thus the other 2 points of infection are

To verify that they are all inflection points we need to plug in values higher and lower than each value and see if the sign changes.

Lets plug in

Since there is a sign change at each point, all are points of inflection.

### Example Question #7 : How To Graph Functions Of Points Of Inflection

Find the points of inflection of

.

There are no points of inflection.

There are no points of inflection.

Explanation:

In order to find the points of inflection, we need to find

Now we set .

.

This last statement says that  will never be . Thus there are no points of inflection.

### Example Question #8 : How To Graph Functions Of Points Of Inflection

Find the points of inflection of the following function:

Explanation:

The points of inflection of a given function are the values at which the second derivative of the function are equal to zero.

The first derivative of the function is

, and the derivative of this function (the second derivative of the original function), is

.

Both derivatives were found using the power rule

Solving .

To verify that this point is a true inflection point we need to plug in a value that is less than the point and one that is greater than the point into the second derivative. If there is a sign change between the two numbers than the point in question is an inflection point.

Lets plug in

.

Now plug in

.

Therefore,  is the only point of inflection of the function.

### Example Question #9 : How To Graph Functions Of Points Of Inflection

Find all the points of inflection of

.

Explanation:

In order to find all the points of inflection, we first find  using the power rule twice, .

Now we set .

.

Now we factor the left hand side.

From this, we see that there is one point of inflection at .

For the point of inflection, lets solve for x for the equation inside the parentheses.

### Example Question #10 : How To Graph Functions Of Points Of Inflection

Find all the points of inflection of:

There are no points of inflection.

Explanation:

In order to find all the points of inflection, we first find  using the power rule twice .

Now we set .

Thus the points of inflection are   and

### Example Question #11 : Points Of Inflection

Which of the following is a point of inflection on ?

Explanation:

Which of the following is a point of inflection on f(x)?

To find points of inflection, we need to find where the second derivative is 0.

So, find f''(x)

So, we have a point of inflection at x=0.

Find f(0) to find the y-coordinate:

So our point is at .

### Example Question #1 : Inflection Points

Which of the following is a point of inflection of  on the interval ?

Explanation:

Which of the following is a point of inflection of f(x) on the interval ?

To find points of inflection, we need to know where the second derivative of the function is equal to zero. So, find the second derivative:

So, where on the given interval does ?

Well, we know from our unit circle that ,

So we would have a point of inflection at , but we still need to find the y-coordinate of our POI. find this by finding

So our POI is:

### Example Question #3 : Inflection Points

Which of the following functions has an inflection point where concavity changes?

Explanation:

For a graph to have an inflection point, the second derivative must be equal to zero. We also want the concavity to change at that point.

, for all real numbers, but this is a line and has no concavity associated with it, so not this one.

, there are no real values of  for which this equals zero, so no inflection points.

, same story here.

, so no inflection points here.

This leaves us with

, whose derivatives are a bit more difficult to take.

, so by the chain rule we get

So,  when . So

.  This is our correct answer.

### Example Question #12 : Points Of Inflection

Find the inflection point of this 3rd degree polynomial:

Explanation:

To find the inflection point we must find where the second derivative of a function is 0.

Calculating the second derivative is fairly simple. We just need to know that :

The first derivative is

and if we take the derivative once more we get

.

Setting this equal to zero, we get

And now all we have to do is plug this value, 3/5, into our original polynomial, to get the answer.

### Example Question #13 : Points Of Inflection

Determine the point(s) of inflection for the function .

This function has no points of inflection.

only

only

and

only