Monotonically Increasing and Decreasing Functions

Monotonicity of the Function is a term that describes how the function goes up or down. Most of the time, the word “monotonic” has two parts. Mono means “one,” and “tonic” means “tone.” Both of these words mean “in the same way.” When you say that a function doesn’t go down, does that mean it goes up? No, that’s not true. It could also mean that the function never changes. In other words, the value of the function stays the same for a certain amount of time. One of the most common ways that derivatives are used is in the increasing-decreasing function. With derivatives, you can find out if a function is going up or down over a certain period.

Monotonic Functions

Monotonicity of the Function is how the function behaves when its increases or decreases.

Any function that follows one of the four cases listed above is called a monotonic function. Most of the time, the word “monotonic” has two parts. Mono means “one,” and “tonic” means “tone.” Both of these words mean “in the same way.” When you say that a function doesn’t go down, does that mean it goes up? No, that’s not true. It could also mean that the Function never changes. In other words, the value of the function stays the same for a certain amount of time. Make sure you don’t confuse not going down with going up.

 Condition for a Monotonic Function

If a function is monotonic, there is a way to tell. This condition has to do with the sign of the function’s derivative.

The derivative of a monotonic function will either always be positive (monotonically going up) or always be negative (monotonically decreasing). In other words, a function is monotonic if its derivative is always positive or negative. Also, a function is called “strictly monotonic” if its derivative is always positive or negative and never zero.

Monotonically Increasing Functions

The graphs of logarithmic and exponential functions will be very important here. We can see a general rule from them:

If an is greater than 1, then both of these functions always go up:

If a>1, then both of these functions are monotonically increasing:

f(x)=ax

g(x)=log a(x)

Take a look at these two graphs. The red one is f(x)=3x and the green one is g(x)=3x+1:

Notice that f(x) gets bigger as x gets bigger. Now, pay attention to this math trick:

g(x)=3x+1=3⋅3x=3f(x)

So, g(x) also keeps going up in the same way. You can multiply f(x) by any positive constant, and as x gets bigger, the values of the new function will keep growing at a faster or slower rate.

Monotonically Decreasing Functions

Functions that always go down are the opposite of functions that always go up. As a result:

If f(x) is a function that always goes up over some time interval, then f(x) is a function that always goes down over the same time interval, and vice versa.

Here is an example of a function that always goes down.

Example: Take a look at the graph of f(x)=5x, which is shown below:

The function 5x always goes up, so the function f(x)=5x must always go down. This is because 5x always goes up, so f(x) must always go down.

When a function is not monotonically increasing or decreasing

 Some functions are neither always going up nor always going down in the same way. There are an infinite number of these functions, and they belong to a lot of different groups.

Main Group 1:  Constant Functions

Since these are straight lines, they don’t go up or down.

Main Group 2: Absolute Value Functions

When a function has an absolute value sign around it, it can never return a negative number. However, all non-constant functions of this type will have a minimum. So, as xx goes up, the function will alternate between going up and going down.

Main Group 3: Trigonometric Functions

Think about basic trigonometric functions like sin(x), which goes up and down and doesn’t just go up or down.

Main Group 4: Functions with Discontinuities

A function can’t go up or down over any kind of break, especially if the break is caused by an undefined value, like when x = 0 in f(x) = 1x.

Graphs can be used to find graphs that go up or down in a steady way, but this is not a very accurate way to do it. Still, it’s a good book for students who don’t know much about calculus. There are more detailed and rigorous methods that use calculus and have to do with the rate of change of the function as x changes.

Increasing and Decreasing Functions: The First Derivative Test

We’ll learn how to use the first derivative test in this section. This is used to figure out how often a function is getting bigger or smaller. Let’s look at the graph of 3x3-3x to help us understand:

The graph can be split into three parts, as you can see. The first is growing; it goes from negative infinity to a peak. Then, from that peak, it goes down until it reaches the bottom of a valley. At the bottom of that valley, it starts to go up again, and it will keep going up until there are no more numbers. Follow these steps to solve them:

1. Take the derivative of the function

2. Find the important numbers (solve for f ‘ (x) = 0)

3. This is what gives us our gaps. Now, pick a number from each of these ranges and plug it into the derivative. If the number is positive, the interval is getting longer. If the number is negative, the interval is getting shorter.

Conclusion

Monotonicity is one of the fundamental principles to understand when applying derivatives to a problem. The monotonicity of a function provides insight into how the function will behave in various situations. If a function’s graph is only rising with increasing values of the equation, then we say that the function has monotonically increasing behaviour.