# 65 5.7 The Second Derivative Test

## Introduction

Welcome to another math tutorial! In this article, we will be diving into the fascinating world of calculus and exploring a concept known as the second derivative test. If you're a math enthusiast or a student looking to deepen your understanding of calculus, you've come to the right place. So, let's get started!

## Understanding the Second Derivative

### What is a derivative?

Before we delve into the second derivative, let's quickly recap what a derivative is. In calculus, a derivative measures how a function changes as its input variable changes. It provides us with valuable information about the rate of change of a function at any given point.

### What is the second derivative?

The second derivative, as the name suggests, is the derivative of a derivative. It captures the rate at which the rate of change of a function is itself changing. In other words, it tells us how the slope of a function is changing at each point on its graph.

## Using the Second Derivative Test

### What is the Second Derivative Test?

The second derivative test is a method used to determine the nature of critical points of a function. It helps us classify these points as local maxima, local minima, or neither.

### The Second Derivative Test: Step-by-Step

Here's how you can utilize the second derivative test to analyze the critical points of a function:

### Step 1: Find the critical points

Begin by finding the critical points of the function. These are the points where the derivative of the function equals zero or is undefined.

### Step 2: Calculate the second derivative

Once you have identified the critical points, calculate the second derivative of the function.

### Step 3: Evaluate the second derivative at the critical points

Next, substitute the critical points into the second derivative and evaluate the resulting values.

### Step 4: Analyze the results

Based on the values obtained in the previous step, you can classify the critical points as follows:

### Case 1: Second derivative > 0

If the value of the second derivative at a critical point is positive, then the function has a local minimum at that point.

### Case 2: Second derivative < 0

If the value of the second derivative at a critical point is negative, then the function has a local maximum at that point.

### Case 3: Second derivative = 0

If the value of the second derivative at a critical point is zero, the test is inconclusive. In this case, you need to resort to additional methods to determine the nature of the critical point.

## Example Application

### Let's work through an example

Now that we have a solid understanding of the second derivative test, let's apply it to a real-world example. Consider the function f(x) = x^3 - 3x^2 - 9x + 5.

### Step 1: Find the critical points

To find the critical points, we need to set the derivative of f(x) equal to zero and solve for x. In this case, f'(x) = 3x^2 - 6x - 9. By factoring or using the quadratic formula, we find that the critical points are x = -1 and x = 3.

### Step 2: Calculate the second derivative

Now, let's find the second derivative f''(x). Differentiating f'(x), we get f''(x) = 6x - 6.

### Step 3: Evaluate the second derivative at the critical points

Substituting x = -1 and x = 3 into f''(x), we find that f''(-1) = -12 and f''(3) = 12.

### Step 4: Analyze the results

Based on the values obtained, we can conclude that the function has a local maximum at x = -1 and a local minimum at x = 3.

## Conclusion

The second derivative test is a powerful tool in calculus that allows us to analyze the behavior of functions at critical points. By calculating the second derivative and evaluating it at these points, we can determine whether they correspond to local maxima, local minima, or inconclusive results. Understanding this test can greatly enhance our ability to analyze and interpret functions in the realm of calculus. So, keep practicing and exploring the wonders of calculus!