Fall 2025: Chapter 4.1 & 4.2 - Finding Extrema

Welcome to a journey through the fascinating world of finding the maximum and minimum values of functions! In Chapters 4.1 and 4.2, we explore how to identify these extrema, which are critical in understanding the behavior of functions. Let's break down the key concepts:

Definitions: Absolute vs. Local Extrema

  • Absolute Maximum: For a function $f$ defined on a domain $D$, $f(c)$ is the absolute maximum value if $f(c) \geq f(x)$ for all $x$ in $D$. In simpler terms, it's the highest point of the function on the given domain.
  • Absolute Minimum: Similarly, $f(c)$ is the absolute minimum value if $f(c) \leq f(x)$ for all $x$ in $D$. This is the lowest point of the function on the given domain.
  • Local Maximum: The value $f(c)$ is a local maximum if $f(c) \geq f(x)$ when $x$ is near $c$. It's a peak in the neighborhood of $c$.
  • Local Minimum: The value $f(c)$ is a local minimum if $f(c) \leq f(x)$ when $x$ is near $c$. It's a valley in the neighborhood of $c$.

Consider the example function: $f(x) = 3x^4 - 16x^3 + 18x^2$ for $-1 \leq x \leq 4$. We can visually identify the local and absolute extrema on the graph.

Theorems: Ensuring Extrema Exist

  • The Extreme Value Theorem: If $f$ is continuous on a closed interval $[a, b]$, then $f$ attains an absolute maximum value $f(c)$ and an absolute minimum value $f(d)$ at some numbers $c$ and $d$ in $[a, b]$. This theorem guarantees the existence of extrema under specific conditions.

Fermat's Theorem and Critical Numbers

  • Fermat's Theorem: If $f$ has a local maximum or minimum at $c$, and if $f'(c)$ exists, then $f'(c) = 0$. This theorem provides a powerful tool for finding local extrema.

A critical number of a function $f$ is a number $c$ in the domain of $f$ such that either $f'(c) = 0$ or $f'(c)$ does not exist. Critical numbers are crucial because, according to Fermat's Theorem, local extrema can only occur at critical numbers.

Finding Critical Numbers: Examples

Let's find the critical numbers for a few example functions:

  1. Consider the function $f(x) = x^3 - 3x^2 - 9x + 4$. Taking the derivative, we get $f'(x) = 3x^2 - 6x - 9$. Setting $f'(x) = 0$, we have $3x^2 - 6x - 9 = 0$. Simplifying, $x^2 - 2x - 3 = 0$, which factors to $(x - 3)(x + 1) = 0$. Thus, the critical numbers are $x = 3$ and $x = -1$.
  2. For $f(x) = x^3$, we have $f'(x) = 3x^2$. Setting $3x^2 = 0$, we find $x = 0$. Therefore, $x=0$ is the critical number.
  3. For $f(x) = |x|$, we can write $f(x) = \begin{cases} -x & x < 0 \\ x & x \geq 0 \end{cases}$. Then, $f'(x) = \begin{cases} -1 & x < 0 \\ 1 & x > 0 \end{cases}$. The derivative does not exist at $x = 0$, so $x = 0$ is a critical number.

Remember to always check the endpoints of the interval when finding absolute extrema!

Keep practicing, and you'll become a master at finding those extrema!