Table of Contents
Maxima and Minima
A maximum is the highest value that a function can attain in a given region.
A minimum is the lowest value that a function can attain in a given region.
Local Maximum:
A local maximum is a point in a function at which the function’s value is greater than all of the function’s values at nearby points.
Local Minimum:
A local minimum is a point in the function where the function takes on a minimum value in a small neighborhood around that point.
The function has a local minimum at x = 2.
The function takes on a minimum value in a small neighborhood around x = 2.
Global (or Absolute) Maximum and Minimum:
The absolute maximum and minimum are the largest and smallest points on a graph.
Finding Maxima and Minima Using Derivatives
Finding maxima and minima of a function is a process of finding the points where the derivative of the function is zero.
The derivative of a function is a measure of how the function changes as you move closer to a point. A function has a maximum at the point where the derivative is zero. The function has a minimum at the point where the derivative is zero.
To find the derivative of a function, you use the derivative function. The derivative function is usually written as .
The derivative of a function can be negative or positive. A function has a maximum when the derivative is positive and a minimum when the derivative is negative.
The following steps can be used to find the maximum and minimum of a function:
Step 1: Find the derivative of the function.
Step 2: Find the points where the derivative is zero.
Step 3: Check the sign of the derivative at each point to determine whether it is a maximum or a minimum.
Step 4: If the derivative is positive at a point, the function has a maximum at that point.
Step 4: If the derivative is negative at a point, the function has a minimum at that point.
How to Calculate Maxima and Minima Points?
To calculate the maximum and minimum points of a graph, one needs to find the derivative of the graph and then set it equal to zero. The maximum and minimum points will be the points where the derivative is equal to zero.
Derivative Tests:
The derivative of a function can be used to determine whether a function is increasing, decreasing, or constant.
There are three derivative tests that can be used to determine the nature of a function’s graph:
1. The First Derivative Test
2. The Second Derivative Test
3. The Concavity Test
1. The First Derivative Test
The First Derivative Test is used to determine whether a function is increasing, decreasing, or constant.
To use the First Derivative Test, you must first find the derivative of the function.
Then, you must determine whether the derivative is positive, negative, or zero.
If the derivative is positive, the function is increasing.
If the derivative is negative, the function is decreasing.
If the derivative is zero, the function is constant.
2. The Second Derivative Test
The Second Derivative Test is used to determine whether a function is concave up or concave down.
To use the Second Derivative Test, you must first find the derivative of the function.
Then, you must determine whether the second derivative is positive or negative.
If the second derivative is positive, the function is concave up.
If the second derivative is negative, the function is concave down.
3. The Concavity Test
The Concavity
First Order Derivative Test:
If is continuous on the interval and derivable on the interval , then there exists a point in the interval at which the derivative is equal to .
Proof:
We first need to show that is continuous on the interval . For this, we need to show that the limit of as approaches is equal to . We can do this by using the definition of a limit:
Since is continuous on the interval and is derivable on the interval , we can use the function theorem to conclude that there exists a point in the interval at which the derivative is equal to .
Second Derivative Test:
If the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down. If the second derivative is zero, the function is at a local minimum. If the second derivative is undefined, the function is at a local maximum.
How to Find Maxima Functions?
There are a few different ways to find maxima functions. One way is to use a graphing calculator to find the points where the function has the largest value. Another way is to use a derivative to find the points where the function has a derivative of zero.
