Maxima/Minima Based Problems - First Derivative Test

Understanding maxima and minima is a fundamental aspect of calculus, particularly in the study of functions and their applications. The first derivative test is a reliable method for identifying the local maximum and minimum points of a function. Here, we will delve into the concept of maxima and minima, the first derivative test, and how to apply it to solve problems.

Understanding Maxima and Minima

Maxima and minima refer to the highest and lowest points, respectively, on a graph of a function within a given region. These points are collectively known as extrema. There are two types of extrema:

• Local Maximum: A point on the graph where the function value is higher than all other values in its immediate vicinity.
• Local Minimum: A point on the graph where the function value is lower than all other values in its immediate vicinity.

Additionally, if the extrema are the highest or lowest points on the entire graph, they are referred to as absolute or global maxima and minima.

First Derivative Test

The first derivative test is a technique used to find the local maxima and minima of a function. It is based on the behavior of the derivative of the function.

Steps for the First Derivative Test

1. Find the derivative: Compute the first derivative $f'(x)$ of the function $f(x)$.
2. Identify critical points: Solve $f'(x) = 0$ to find the critical points. These are potential candidates for local maxima or minima.
3. Analyze the sign of $f'(x)$: Determine the sign of the derivative before and after each critical point.
4. Conclude: If $f'(x)$ changes from positive to negative at a critical point, $x=c$, then $f(c)$ is a local maximum. If $f'(x)$ changes from negative to positive at $x=c$, then $f(c)$ is a local minimum.

Table of Differences and Important Points

Property	Local Maximum	Local Minimum
Definition	Function value at this point is higher than nearby points.	Function value at this point is lower than nearby points.
First Derivative Test	$f'(x)$ changes from positive to negative.	$f'(x)$ changes from negative to positive.
Graph Behavior	The graph of $f(x)$ peaks at this point.	The graph of $f(x)$ troughs at this point.

Formulas

The first derivative test doesn't have a specific formula, but it relies on the computation of the first derivative:

$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$

Examples

Let's go through a couple of examples to illustrate the first derivative test.

Example 1: Quadratic Function

Consider the function $f(x) = x^2 - 4x + 3$.

1. Find the derivative: $f'(x) = 2x - 4$.
2. Identify critical points: Solve $f'(x) = 0 \Rightarrow 2x - 4 = 0 \Rightarrow x = 2$.
3. Analyze the sign of $f'(x)$: For $x < 2$, $f'(x) > 0$; for $x > 2$, $f'(x) < 0$.
4. Conclude: Since $f'(x)$ changes from positive to negative at $x=2$, $f(2) = -1$ is a local maximum.

Example 2: Cubic Function

Consider the function $f(x) = x^3 - 3x^2 - 9x + 10$.

1. Find the derivative: $f'(x) = 3x^2 - 6x - 9$.
2. Identify critical points: Solve $f'(x) = 0 \Rightarrow 3x^2 - 6x - 9 = 0$. This gives us $x = -1$ and $x = 3$.
3. Analyze the sign of $f'(x)$: For $x < -1$, $f'(x) > 0$; for $-1 < x < 3$, $f'(x) < 0$; for $x > 3$, $f'(x) > 0$.
4. Conclude: Since $f'(x)$ changes from positive to negative at $x=-1$, $f(-1) = 23$ is a local maximum. Since $f'(x)$ changes from negative to positive at $x=3$, $f(3) = -2$ is a local minimum.

By following these steps and understanding the behavior of the derivative, one can effectively use the first derivative test to find local maxima and minima in various functions, which is a crucial skill in calculus and its applications.