First Derivative Test: Increasing/Decreasing Functions and Local Extrema

Increasing and Decreasing Functions

The connection between the derivative and a function's direction is very intuitive: if the slope of the tangent line is positive, the function is rising; if the slope is negative, it's falling. The Mean Value Theorem provides the rigorous mathematical justification for this intuition.

The theorem you just read can be summarized as follows for a function $f$ on an interval $I$:

• If $f^{\prime }(x)>0$ for all $x$ in $I$, then $f$ is increasing on $I$.
• If $f^{\prime }(x)<0$ for all $x$ in $I$, then $f$ is decreasing on $I$.

This gives us a clear plan for finding where a function increases or decreases: we need to find the intervals where its derivative is positive or negative. A function can only change from increasing to decreasing (or vice versa) where its derivative is zero or undefined. This brings us back to a familiar concept: critical points.

The procedure is as follows:

1. Find the derivative of the function, $f^{\prime }(x)$.
2. Find the critical points of $f$ by determining where $f^{\prime }(x)=0$ or where $f^{\prime }(x)$ is undefined.
3. Use the critical points to divide the number line into test intervals.
4. Choose a test value from each interval and plug it into the derivative, $f^{\prime }(x)$, to determine its sign.
5. Use the sign of $f^{\prime }(x)$ to determine if $f(x)$ is increasing or decreasing on that interval.

Let's see this process in action.

The First Derivative Test

Now we can take the final step. If we know where a function is increasing and decreasing, we can identify its local maximum and minimum values.

Think about a smooth curve.

• If the function changes from increasing to decreasing at a critical point $c$, it must form a peak. This is a local maximum.
• If the function changes from decreasing to increasing at a critical point $c$, it must form a valley. This is a local minimum.
• If the function's derivative does not change sign at $c$ (for example, it's increasing on both sides), then $c$ is neither a local maximum nor a minimum.

This logical conclusion is called the First Derivative Test.

Let's formalize this.

The process for finding and classifying local extrema is an extension of our previous one:

1. Find the derivative $f^{\prime }(x)$.
2. Find the critical points.
3. Create a sign chart (number line) for $f^{\prime }(x)$.
4. Determine intervals of increase and decrease.
5. Apply the First Derivative Test at each critical point to classify it as a local maximum, local minimum, or neither.

A Comprehensive Example

Let's work through a more involved example that showcases the power and nuances of this test, especially when dealing with critical points where the derivative is undefined.

Consider the function $f(x)=3x^{5/3}-15x^{2/3}$. We want to find its intervals of increase and decrease and classify all local extrema.

Step 1: Find the derivative.
Using the power rule:

Step 2: Find the critical points.
To make this easier, let's factor the derivative. The common factor is $5x^{-1/3}$.

Critical points occur where $f^{\prime }(x)=0$ or is undefined.

• $f^{\prime }(x)=0$ when the numerator is zero: $5(x-2)=0⟹x=2$.
• $f^{\prime }(x)$ is undefined when the denominator is zero: $x^{1/3}=0⟹x=0$.

So, our critical points are $x=0$ and $x=2$.

Step 3 & 4: Create a sign chart and test intervals.
We divide the number line into three intervals: $(-\infty ,0)$, $(0,2)$, and $(2,\infty )$.

Step 5: Apply the First Derivative Test.

• At $x=0$, the derivative changes from positive to negative. Therefore, $f(x)$ has a local maximum at $x=0$. The value is $f(0)=0$.
• At $x=2$, the derivative changes from negative to positive. Therefore, $f(x)$ has a local minimum at $x=2$. The value is $f(2)=3(2{)}^{5/3}-15(2{)}^{2/3}\approx -14.3$.

This example is important because it shows that you must consider points where the derivative is undefined. The function has a sharp corner, or "cusp," at $x=0$, but this point is still a local maximum.

Professor Leonard provides an excellent walkthrough of a very similar problem. Watching him solve it will help solidify this process.

Conclusion

In this lesson, we have developed one of the most practical tools in differential calculus. By analyzing the sign of the first derivative, we can now confidently map out the behavior of a function.

Key Takeaways:

• The sign of the first derivative tells you about the direction of the function: $f^{\prime }(x)>0$ means $f(x)$ is increasing, and $f^{\prime }(x)<0$ means $f(x)$ is decreasing.
• Critical points (where $f^{\prime }(x)=0$ or is undefined) are the only places where a function can change from increasing to decreasing or vice versa.
• The First Derivative Test formalizes this analysis:
  • A sign change from + to - in $f^{\prime }(x)$ indicates a local maximum.
  • A sign change from - to + in $f^{\prime }(x)$ indicates a local minimum.
  • No sign change means the critical point is not a local extremum.

This skill is the first major step toward being able to sketch a detailed graph of a function without relying on a calculator. However, our sketches are still missing a key piece of information: the curvature or concavity of the graph. In our next lesson, we will introduce the second derivative and the Second Derivative Test, which will allow us to determine concavity and provide an alternative method for classifying extrema.