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# Justification using first derivative

Let's take a close look at how the behavior of a function is related to the behavior of its derivative. This type of reasoning is called "calculus-based reasoning." Learn how to apply it appropriately.
A derivative ${f}^{\prime }$ gives us all sorts of interesting information about the original function $f$. Let's take a look.

## How ${f}^{\prime }$‍  tells us where $f$‍  is increasing and decreasing

Recall that a function is increasing when, as the $x$-values increase, the function values also increase.
Graphically, this means that as we go to the right, the graph moves upwards. Similarly, a decreasing function moves downwards as we go to the right.
Now suppose we don't have the graph of $f$, but we do have the graph of its derivative, ${f}^{\prime }$.
We can still tell when $f$ increases or decreases, based on the sign of the derivative ${f}^{\prime }$:
• The intervals where the derivative ${f}^{\prime }$ is $\text{positive}$ (i.e. above the $x$-axis) are the intervals where the function $f$ is $\text{increasing}$.
• The intervals where ${f}^{\prime }$ is $\text{negative}$ (i.e. below the $x$-axis) are the intervals where $f$ is $\text{decreasing}$.
When we justify the properties of a function based on its derivative, we are using calculus-based reasoning.
Problem 1
These are two valid justifications for why a function $f$ is an increasing function:
$A$. As the $x$-values increase, the values of $f$ also increase.
$B$. The derivative of $f$ is always positive.
Which of the above is a calculus-based justification?

Problem 2
The differentiable function $f$ and its derivative ${f}^{\prime }$ are graphed.
What is an appropriate calculus-based justification for the fact that $f$ is decreasing when $x>3$?

#### Common mistake: Not relating the graph of the derivative and its sign.

When working with the graph of the derivative, it's important to remember that these two facts are equivalent:
• ${f}^{\prime }\left(x\right)<0$ at a certain point or interval.
• The graph of ${f}^{\prime }$ is below the $x$-axis at that point/interval.
(The same goes for ${f}^{\prime }\left(x\right)>0$ and being above the $x$-axis.)

## How ${f}^{\prime }$‍  tells us where $f$‍  has a relative minimum or maximum

In order for a function $f$ to have a relative maximum at a certain point, it must increase before that point and decrease after that point.
At the maximum point itself, the function is neither increasing nor decreasing.
In the graph of the derivative ${f}^{\prime }$, this means that the graph crosses the $x$-axis at the point, so the graph is above the $x$-axis before the point and below the $x$-axis after.
Problem 3
The differentiable function $g$ and its derivative ${g}^{\prime }$ are graphed.
What is an appropriate calculus-based justification for the fact that $g$ has a relative minimum point at $x=-3$?

#### Common mistake: Confusing the relationship between the function and its derivative

As we saw, the sign of the derivative corresponds to the direction of the function. However, we can't make any justification based on any other kinds of behavior.
For example, the fact that the derivative is increasing doesn't mean the function is increasing (or positive). Furthermore, the fact that the derivative has a relative maximum or minimum at a certain $x$-value doesn't mean the function must have a relative maximum or minimum at that $x$-value.
Problem 4
The differentiable function $h$ and its derivative ${h}^{\prime }$ are graphed.
Four students were asked to give an appropriate calculus-based justification for the fact that $h$ is increasing when $x>0$.
Can you match the teacher's comments to the justifications?

Want more practice? Try this exercise.

#### Common mistake: Using obscure or non-specific language.

There are a lot of factors at play when we’re looking at the relationship between a function and its derivative: the function itself, that function’s derivative, the direction of the function, the sign of the derivative, etc. It's important to be extremely clear about what one is talking about at any given time.
For example, in Problem 4 above, the correct calculus-based justification for the fact that $h$ increasing is that ${h}^{\prime }$ is positive, or above the $x$-axis. One of the students' justifications was "It's above the $x$-axis." The justification didn't specify what is above the $x$-axis: the graph of $h$? The graph of ${h}^{\prime }$? Or maybe something else? Without being specific, such a justification cannot be accepted.

## Want to join the conversation?

• is there any use to knowing the justification?