# Calculus 1 : Decreasing Intervals

## Example Questions

### Example Question #1571 : Functions

For which values of  is the function decreasing?

and

Explanation:

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

### Example Question #1572 : Functions

The picture below shows the graph of some function,

On which interval of  is the function decreasing?

Interval C

Interval A

Interval D

Interval B

Interval E

Interval D

Explanation:

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.

### Example Question #1573 : Functions

Find the intervals on which the function is decreasing:

The function is never decreasing.

Explanation:

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.

### Example Question #11 : Decreasing Intervals

Given , find the interval over which  is decreasing.

The function is always increasing.

Explanation:

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 .

### Example Question #12 : Decreasing Intervals

Determine the intervals on which the given function is decreasing:

The function is never decreasing

Explanation:

To determine the intervals on which the function is decreasing, we must find the intervals on which the function's first derivative is negative. To do this, we must find the first derivative and the critical value(s) at which the first derivative is equal to zero:

The derivative was found using the following rule:

Now, setting the first derivative equal to zero, we get

So, now we can make our intervals to be analyzed (is the first derivative positive or negative on the interval?), in which c is the upper and lower bound:

Note that at c the first derivative is neither positive nor negative.

On the first interval, the first derivative is always negative, so the function is always decreasing on this interval. On the second interval, the first derivative is always positive, therefore the function is increasing on this interval.

We are concerned with the interval where the function is decreasing, so  is our answer.

### Example Question #13 : Decreasing Intervals

Find the intervals on which the given function is decreasing:

Explanation:

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

First, we must find the first derivative:

The first derivative was found using the following rule:

Now, we set the first derivative equal to zero and solve for the critical values - values at which the first derivative is equal to zero:

Now, using the critical values, we create the intervals in which we see whether the first derivative is positive or negative:

On the first interval, the derivative is positive, on the second it is negative, and on the third interval it is positive (we simply plug in any point in the interval into the first derivative function and check the sign).

So, the function is decreasing on the second interval .

### Example Question #14 : Decreasing Intervals

Find the interval(s) on which the function is decreasing:

The function is never decreasing

The function is never decreasing

Explanation:

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.

First, we find the first derivative:

It was found using the following rule:

Now, we must find the critical values - values at which the first derivative is equal to zero:

Now, using the critical values as upper and lower bounds, we create the intervals on which we determine the sign of the first derivative:

On the first interval, the first derivative is always positive, and on the second interval, the first derivative is always positive. (Simply plug in any point in the interval into the first derivative function and check the sign.) Thus, the function is never decreasing.

### Example Question #15 : Decreasing Intervals

Find the interval(s) where the function  is decreasing.

Explanation:

To find the intervals where the function decreases, apply the first derivative test. Find the derivative, set equal to , and solve to find local extrema.

So  or .

Next, test points in each of the intervals delineated by the potential local extrema. For example:

, so the function increases to the left of .

, so the function decreases on the interval

, so the function increases to the right of .

### Example Question #16 : Decreasing Intervals

Find the interval in which the following function is decreasing.

never

always

never

Explanation:

To find decreasing intervals, you must find when the first derivative is less than 0. Differentiate using the power rule:

Thus,

Since 2 is never negative, our first derivative is never negative. Therefore, our function is never decreasing.

### Example Question #17 : Decreasing Intervals

Find the intervals on which the function is decreasing:

The function is never decreasing

Explanation:

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

So, the first derivative of the function is equal to

and was found using the following rule:

Next, we must find the critical values, at which the first derivative is equal to zero:

Now, we make the intervals, using c as our upper and lower bound:

To determine whether the first derivative is positive or negative each interval, simply plug in any number on the interval into the first derivative function. On the first interval, the first derivative is positive, while on the second interval, the first derivative is negative. Our answer is therefore .