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### Course: AP®︎/College Calculus AB > Unit 5

Lesson 10: Connecting a function, its first derivative, and its second derivative- Calculus-based justification for function increasing
- Justification using first derivative
- Justification using first derivative
- Justification using first derivative
- Inflection points from graphs of function & derivatives
- Justification using second derivative: inflection point
- Justification using second derivative: maximum point
- Justification using second derivative
- Justification using second derivative
- Connecting f, f', and f'' graphically
- Connecting f, f', and f'' graphically (another example)
- Connecting f, f', and f'' graphically

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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 ${f}^{\prime}$ .

*derivative*,We can still tell when $f$ increases or decreases, based on the ${f}^{\prime}$ :

*sign*of the derivative- The intervals where the derivative
is${f}^{\prime}$ (i.e. above the${\text{positive}}$ -axis) are the intervals where the function$x$ is$f$ .${\text{increasing}}$ - The intervals where
is${f}^{\prime}$ (i.e. below the${\text{negative}}$ -axis) are the intervals where$x$ is$f$ .${\text{decreasing}}$

When we justify the properties of a function based on its derivative, we are using

**calculus-based**reasoning.#### 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:

at a certain point or interval.${f}^{\prime}(x)<0$ - The graph of
is below the${f}^{\prime}$ -axis at that point/interval.$x$

(The same goes for ${f}^{\prime}(x)>0$ 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 ${f}^{\prime}$ , this means that the graph $x$ -axis before the point and below the $x$ -axis after.

*derivative**crosses the*$x$ -axisat the point, so the graph is above the#### 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.

*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 "$x$ -axis." The justification didn't specify $x$ -axis: the graph of $h$ ? The graph of ${h}^{\prime}$ ? Or maybe something else? Without being specific, such a justification cannot be accepted.

*above the***It's***is above the***what**## Want to join the conversation?

- is there any use to knowing the justification?(3 votes)
- To past the AP EXAM😇(47 votes)

- What does it mean when f' of a certain value doesn't exist?(2 votes)
- If f’ does not exist at a certain point, then the function is not differentiable at that point. This could mean that there is a discontinuity at that point or maybe there is a “cusp” or sharp turn in the graph.(1 vote)

- I do not understand this statement "For example, the fact that the derivative is increasing doesn't mean the function is increasing (or positive)."(1 vote)
- That other answer is incorrect. As per the quoted statement, there is no direct causation between the derivative increasing and either the function, itself, being positive OR increasing. Recall that the derivative is the slope at any given instant/point (ie, where the function is going): essentially that quote is stating that where the function is going is a separate matter from where the function is currently at. Say a continuous and differentiable function is approaching a local minimum between outputs f(x) = -1 and f(x) = -3; somewhere therein the function's derivative would be increasing (as the function as about to turn around -- ie: attain a derivative of 0,) while the function itself is both negative (below the x-axis,) AND decreasing (approaching the local minimum.)(3 votes)