Functions can either be **increasing** or **decreasing** for different intervals. And without looking at a graph of the function, you can't tell visually what a function is doing.

Fortunately, in business calculus, we can use derivatives to determine when a function is increasing or decreasing over a determined interval.

Below, we will cover increasing and decreasing functions in business calculus, including what they are, and how to determine whether a function is increasing or decreasing when a graph is available (using your fingers), and when a graph is not available (using derivatives).

When a graph is available, a function is increasing if the function is rising when looked at from left to right.

A function is increasing for a certain interval if, when traced with a finger from left to right over that interval, the function is going up (in other words, its $y$ value is increasing)

Basically, for an increasing function, as the values of $x$ (input) increase, the values of $y$ (output) also increase.

By the way, you might be wondering what "tracing a finger" through the graph of the function means:

When a graph is available, there is an easy way to determine whether a graph is increasing or decreasing: run your finger along the graph from left to right. If your finger goes up as you move to the right, the graph is increasing. If it goes down instead, the graph is decreasing.

Going back to increasing functions, let's look at an example of a function that is always increasing:

$$f(x)=x^3$$

This is a typical cubic function. To see how this function is increasing, click here to view its graph.

Now, go back to the graph and trace your finger through the graph, moving from left to right. As you trace your finger, you will notice the graph goes up (increases) as you move to the right. Because the graph is always rising, we say that this function is always increasing.

The opposite of an increasing function, a function is decreasing if the function is declining when looked at from left to right.

A function is decreasing for a certain interval if, when traced with a finger from left to right over the interval, the function is going down (in other words, its $y$ value is decreasing)

Put another way, for a decreasing function, as the values of $x$ increase (input), the values of $y$ (ouput) decrease instead.

Below is an example of a function that is always decreasing:

$$f(x)=-x+3$$

To see how it is always decreasing, click here to view its graph.

Doing the same test of tracing your finger through the graph from left to right, you will notice that this graph is always going down, and therefore, is always decreasing.

Some functions are neither increasing nor decreasing for a certain interval. We say that the function is constant for that interval. And the derivative of a constant function is $0$, as per the Constant Rule

In the two examples above, one function is always increasing, and the other is always decreasing. However, many functions increase for some values of $x$, and then decrease for other values of $x$. In other words, they are increasing for some intervals, and decreasing for other intervals. An example of this is the sine function:

$$f(x)=sin(x)$$

To view how this function both increases and decreases, click here to view its graph.

Starting from the left, you see that the function is decreasing until about $x = -1.5$. After that, it increases until about $x = 1.5$, and then it starts decreasing again.

Many times, you will need to determine when a function is increasing or decreasing, but the graph is not available. In those cases, you need to use derivatives to determine this.

In words: If the derivative of a function is positive for a certain interval, then that function is increasing over that interval. If the derivative is negative instead, then the function is decreasing over the interval.

In math notation: If $f'(x) > 0$ for an interval, then $f(x)$ is increasing on that interval. If $f'(x) < 0$ for an interval instead, then $f(x)$ is decreasing on that interval.

- Step 1: Find the derivative of the function for which you are you are trying to determine increasing and decreasing intervals
- Step 2: Find the values which make the derivative equal to $0$
- Step 3: Test values on both sides of the zeroes by plugging into the derivative function with the help of a table. If the result is positive, the original function is increasing on the interval. If the result is negative, the original function is decreasing on the interval. We need to check this for all possible intervals

As explained above, you need to determine where the derivative is positive and where it is negative in order to determine when the function is increasing and decreasing. Below are the steps to find when a function is increasing or decreasing using its derivative:

To find where a function is increasing or decreasing, find the derivative of the function and find the values that make the derivative equal to zero. Then, select values to the left and to the right of the zeroes and plug them into the derivative, and check the sign of each result

Given the function $f(x)=4x^2-56x+10$,

Question 1Determine where this function is increasing and where it is decreasing

We will use derivatives to find out when this function is increasing or decreasing. Step 1 says to find the derivative of $f(x)$,

$$f'(x)=8x-56$$

Now that we have the derivative, we move on to Step 2, where we find the values that make the derivative $0$. To do that, we set $f'(x)=0$ and solve for $x$,

$$8x-56=0$$

We add 56 on both sides,

$$8x=56$$

And we divided by 8 on both sides to get our result,

$$x=7$$

This tells us that there is a zero when $x=7$. For Step 3, we need to test any value both to the left (less than) and to the right (greater than) of $7$ by plugging them into the derivative, which we can use a table for.

$x$ | $f'(x)$ | $+$ or $-$ |
---|---|---|

$3$ | $8(3)-56=-32$ | $-$ |

$10$ | $8(10)-56=24$ | $+$ |

As you can see, we picked $3$ and $10$ to test values to the left and right of $7$. In theory, you can pick any numbers you want to test, but these two were easy to plug in.

When picking numbers to test to the left and right of zeroes (numbers that make the derivative equal to $0$), in theory you can pick any $2$ numbers. But why make it complicated? Pick values that are close to the zeroes and that are easy to calculate, such as $0$, $-2$ and $10$. Of course, you can't pick the zeros themselves.

From looking at the table, you will notice that after plugging in these numbers in to the derivative, $3$ gave a negative result and $10$ gave a positive result.

This means that we have a final answer with two intervals:

$in \ (-\infty,7) \ \rightarrow \ f(x) \ is \ decreasing$

$in \ (7,+\infty) \ \rightarrow \ f(x) \ is \ increasing$

To put this in words, it means that the function is decreasing up until 7, after which it starts increasing. We are done.

By the way, it's important to note that you need to test each interval possible for these problems. In the example above, there were only two intervals: to the left of $7$ and to the right of $7$.

However, if for example, the derivative has zeroes at $-2, 4$ and $8$, and the domain of the function is all numbers, then you need to test the following intervals:

- $(-\infty,-2)$, to the left of $-2$
- $(-2,4)$, between $-2$ and $4$
- $(4,8)$, between $4$ and $8$
- $(8,+\infty)$

- If you have the graph of a function, you can determine if the function is increasing or decreasing by running your finger through the graph, from left to right
- If the function is going up, it is increasing. Otherwise, if the function is going down, it is decreasing
- When a graph is not available, you need to find the values that make the derivative equal to $0$, and test values on both sides of the zeroes
- If the derivative if positive for that interval, the original function is increasing there. If the derivative is negative instead, the original function is decreasing there

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