Fall 2025 Chapter 4.3: Unveiling Function Behavior with Calculus

Welcome to Chapter 4.3! In this section, we'll delve into how calculus can help us understand the behavior of functions, specifically where they are increasing, decreasing, and how to find local maximum and minimum values. Get ready to sharpen your calculus skills!

Increasing/Decreasing Test

The sign of the first derivative, $f'(x)$, gives us a powerful insight into whether a function is increasing or decreasing. Here's the core concept:

  • If $f'(x) > 0$ on an interval, then $f$ is increasing on that interval.
  • If $f'(x) < 0$ on an interval, then $f$ is decreasing on that interval.

In essence, a positive derivative means the function's value is going up as $x$ increases, and a negative derivative means it's going down!

Finding Critical Numbers

Before we can apply the increasing/decreasing test, we need to find the critical numbers of the function. Critical numbers occur where $f'(x) = 0$ or where $f'(x)$ is undefined. These points are potential locations for local maxima or minima.

For example, consider the function $f(x) = \frac{6x}{x^2 + 9}$. To find where this function is increasing or decreasing, we follow these steps:

  1. Find the first derivative, $f'(x)$. Using the quotient rule, we have: $$f'(x) = \frac{6(x^2 + 9) - 6x(2x)}{(x^2 + 9)^2} = \frac{-6x^2 + 54}{(x^2 + 9)^2}$$
  2. Find the critical numbers by setting $f'(x) = 0$: $$\frac{-6x^2 + 54}{(x^2 + 9)^2} = 0$$ This simplifies to $-6x^2 + 54 = 0$, so $x^2 = 9$, and $x = \pm 3$.
  3. Create a sign chart: Test values in the intervals $(-\infty, -3)$, $(-3, 3)$, and $(3, \infty)$ to determine the sign of $f'(x)$ in each interval.
  4. Based on the sign chart:
    • $f(x)$ is increasing on the interval $(-3, 3)$.
    • $f(x)$ is decreasing on the intervals $(-\infty, -3)$ and $(3, \infty)$.

The First Derivative Test

The First Derivative Test helps us classify critical points as local maxima or local minima:

  • If $f'$ changes from positive to negative at $c$, then $f$ has a local maximum at $c$.
  • If $f'$ changes from negative to positive at $c$, then $f$ has a local minimum at $c$.
  • If $f'$ does not change sign at $c$, then $f$ has no local maximum or minimum at $c$.

Concavity and the Second Derivative

The second derivative, $f''(x)$, tells us about the concavity of the function's graph:

  • If $f''(x) > 0$ on an interval, then the graph of $f$ is concave upward on that interval. (Think: like a cup holding water)
  • If $f''(x) < 0$ on an interval, then the graph of $f$ is concave downward on that interval. (Think: like a frown)

A point where the concavity changes is called an inflection point.

The Second Derivative Test

The Second Derivative Test provides another way to find local extrema:

  • If $f'(c) = 0$ and $f''(c) > 0$, then $f$ has a local minimum at $c$.
  • If $f'(c) = 0$ and $f''(c) < 0$, then $f$ has a local maximum at $c$.

However, if $f''(c) = 0$, the test is inconclusive.

Understanding increasing and decreasing intervals, concavity, and the first and second derivative tests are fundamental tools in analyzing the behavior of functions. Keep practicing, and you'll master these concepts in no time!