To find the an increasing or decreasing interval, we need to find out if the first derivative is positive or negative on the given interval. So, find  by decreasing each exponent by one and multiplying by the original number.

Next, we can find  and  and see if they are positive or negative.

Both are negative, so the slope of the line tangent to  is negative, so  is decreasing.

Recall that a function is increasing at a point if its first derivative is positive, and a function is decreasing if its first derivative is negative at that point. Therfore, we should start by finding f'(x). However, I will start by combining like terms and putting f(x) in standard form:

Next, plug in each of our endpoints to see what the sign of f'(x) is.

So f'(x) is positive on the given interval, so we know that f(x) is increasing on the given interval.

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.