The first step is to find the first derivative.

We can factor a  out to get

.

Now we need to solve for when  to get the critical points. Notice how factoring the 2 made the expression a little easier to simplify.

The final step is to try points in all the regions  to see which range gives a negative value for .

If we plugin in a number from the first range into , i.e , we get a positive number.

From the second range, , we get a negative number.

From the third range, , we get a positive number.

So the second range gives us values where the function is decreasing because  is negative during that range, so  is the answer.

A function  is strictly decreasing on an inverval if, for any  in the interval,  (i.e the slope is always less than zero)

Interval E is the only interval on which the function shows this property.

 is decreasing on intervals where .

First, differentiate .

Then, find the values for x for which the derivative is negative by solving 

.

Next, test the intervals. 

Test them by substituting values for x:

Substitute -2,

.

The function is increasing on this interval since the derivative is positive on this interval.

Substitute 0,

.

The function is deecreasing on this interval since the derivative is negative on this interval.

Substitute 2,

.

The function is increasing on this interval since the derivative is positive on this interval.

Thus,  is the only interval on which the function is decreasing.

The function is decreasing when the first derivative is negative. We first find when the derivative is zero. To find the derivative, we apply the quotient rule, 

.

Therefore the derivative is zero at . To find when it is negative plug in test points on each of the three intervals created by these zeros.

For instance, 

.

Hence the function is decreasing on 

.

To determine where the function is decreasing, differentiate it:

What we are interested in are the points where . To determine these points, factor the equation:

this has solutions at

This splits the graph into 4 regions, and we can test points in each to determine if is greater than or less than 0. If it is less than zero, the function is decreasing.

 negative/decreasing

 positive/increasing

negative/decreasing

positive/increasing

The function is decreasing where . To determine where this is happening, differentiate the function and find where . This will split the function into intervals where it is either increasing or decreasing.

To determine where this equals zero, factor:

this has solutions for .

Test a point in each region to determine if it is increasing or decreasing within these bounds:

positive/increasing

negative/decreasing

negative/decreasing

positive/increasing

The function is increasing where . To determine the regions where this is true, first take the derivative of :

.

To figure out where this is less than zero, factor and set it equal to zero. This will split the function into intervals where we can test points.

This has solutions at .

Test a point in each region to see if the function is increasing or decreasing:

positive/increasing

negative/decreasing

positive/increasing

A function is decreasing when it has a negative slope. Graphically, this is a region of the curve where the curve decreases as  increases.

On interval D, the curve shows this trait. The curve does not show this trait on any other interval.

Therefore, interval D is the answer. 

To determine the intervals on which the function is decreasing, we must find the intervals on which the first derivative of the function is negative. 

The first derivative of the function is

and we used the following rule:

Now, we find the point(s) at which the first derivative equals zero - the critical value(s):

Now, we make our intervals on which we see whether the first derivative is positive or negative:

On the first interval, the first derivative is negative, while on the second interval it is positive. Thus, the first interval is the one where the function is decreasing. 

To find when a function is decreasing, we must first find where the critical points of the function are. Since we are given the derivative , we start by first setting the derivative equal to  and solving for .

This is our critical point. To evaluate where the function is decreasing, we must check the sign of the derivative on both sides of the critical point. Because , we can check the left side of the critical point by plugging  into .

Because , the function is decreasing on the left side of the critical point, on the interval .

Now we must check the right side of the critical point. Because , we can check the right side of the critical point by plugging  into .

Because , the function is increasing on the right side of the critical point, meaning the only interval on which the function is decreasing is .