# Precalculus : Find the Critical Numbers of a Function

## Example Questions

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### Example Question #1 : Find The Critical Numbers Of A Function

What are the critical  values of the function ?

Explanation:

A number is critical if it makes the derivative of the expression equal 0.

Therefore, we need to take the derivative of the expression and set it to 0. We can use the power rule for each term of the expression.

Next, we need to factor the expression:

We can now set each term equal to 0 to find the critical numbers:

Therfore, our critical numbers are,

### Example Question #2 : Find The Critical Numbers Of A Function

Find the critical value(s) of the function.

Explanation:

To find the critical values of a function, we must set the derivate equal to 0.

First, we find the derivative of the function to be

We can then factor out a 6x and set the expression equal to 0

From here, we can easily determine that

and

.

Therefore, the critical values of the function are at  and .

### Example Question #3 : Find The Critical Numbers Of A Function

Find the critical points of the following function:

Explanation:

The critical points of a function are the points at which its slope is zero, so first we must take the derivative of the function so we have a function that describes its slope:

Now that we have the derivative, which tells us the slope of f(x) at any point x, we can set it equal to 0 and solve for x to find the points at which the slope of the function is 0, which are our critical points:

### Example Question #4 : Find The Critical Numbers Of A Function

Find the critical number(s) of the function

.

DNE

Explanation:

To find the critical numbers, find the values for x where the first derivative is 0 or undefined.

For the function

The first derivative is

So for

the first derivative is

Setting that equal to zero

We find

### Example Question #5 : Find The Critical Numbers Of A Function

Find the critical number(s) of the function

.

DNE

Explanation:

To find the critical numbers, find the values for x where the first derivative is 0 or undefined.

For the function

.

The first derivative is

.

So for

the first derivative is

.

Settig the first derivative equal to zero

yields the values

### Example Question #6 : Find The Critical Numbers Of A Function

Find the critical numbers of the following function:

Explanation:

The critical numbers of a function are those at which its first derivative is equal to 0. These points tell where the slope of the function is 0, which lets us know where the minimums and maximums of the function are. First we find the derivative of the function, then we set it equal to 0 and solve for the critical numbers:

### Example Question #7 : Find The Critical Numbers Of A Function

Find the critical numbers of the following function:

Explanation:

The critical numbers of a function are those at which its first derivative is equal to 0. These points tell where the slope of the function is 0, which lets us know where the minimums and maximums of the function are. First we find the derivative of the function, then we set it equal to 0 and solve for the critical numbers:

### Example Question #8 : Find The Critical Numbers Of A Function

Find the critical numbers of the following function:

Explanation:

The critical numbers of a function are those at which its first derivative is equal to 0. These points tell where the slope of the function is 0, which lets us know where the minimums and maximums of the function are. First we find the derivative of the function, then we set it equal to 0 and solve for the critical numbers:

### Example Question #9 : Find The Critical Numbers Of A Function

Find the critical value(s) of the function .

Explanation:

The critical values of a function  are values for which the derivative . In this case:

Setting :

### Example Question #10 : Find The Critical Numbers Of A Function

Find the critical value(s) of the function .