10 Ways to Find the X Minimum in Desmos

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Are you tired of manually searching through countless data points to find the minimum value? Desmos, the popular online graphing calculator, offers a powerful solution to streamline this process. With its advanced mathematical capabilities, Desmos allows you to effortlessly find the x-minimum of any function, saving you time and effort. In this article, we will guide you through the step-by-step process of using Desmos to determine the x-minimum of any given function.

To begin, you will need to enter the function into Desmos. Once the function is entered, Desmos will generate a graphical representation of the function. The x-minimum of a function is the x-value at which the function reaches its lowest point. To find the x-minimum using Desmos, we can use the “Minimum” tool. This tool allows us to find the minimum value of a function within a specified interval. By adjusting the interval, we can pinpoint the exact x-value of the minimum.

In addition to the “Minimum” tool, Desmos also provides other helpful features for finding the x-minimum. For instance, the “Table” tool can be used to generate a table of values for the function. This table can be used to identify the x-value at which the function reaches its minimum. Furthermore, the “Derivative” tool can be used to find the derivative of the function. The derivative of a function is a measure of its rate of change. By finding the derivative, we can determine the slope of the function at any given point. The x-minimum of a function occurs at a point where the slope of the function is zero.

Introduction to Finding the X Minimum in Desmos

Desmos is a free online graphing calculator that allows users to plot functions, analyze data, and create interactive visualizations. One of the many features that Desmos offers is the ability to find the x-minimum of a function, which is the x-coordinate of the point where the function reaches its lowest value.

There are several ways to find the x-minimum of a function in Desmos, but the most common method is to use the “minimum” function. The minimum function takes a function as its input and returns the x-coordinate of the point where the function reaches its lowest value. For example, to find the x-minimum of the function f(x) = x^2, you would enter the following into Desmos:

“`
minimum(f(x))
“`

Desmos would then return the x-coordinate of the point where f(x) reaches its lowest value, which is 0.

In addition to the minimum function, Desmos also offers several other functions that can be used to find the x-minimum of a function. These functions include the “globalMinimum” function, the “localMinimum” function, and the “extremeValues” function. The globalMinimum function returns the x-coordinate of the point where the function reaches its lowest value over its entire domain, while the localMinimum function returns the x-coordinate of the point where the function reaches its lowest value within a specified interval. The extremeValues function returns the x-coordinates of all the points where the function reaches either its maximum or minimum value.

The following table summarizes the different functions that can be used to find the x-minimum of a function in Desmos:

Using the Minimum Function

The Minimum() function in Desmos finds the minimum value of a given expression over a specified interval. The syntax of the Minimum() function is as follows:

Minimum(expression, variable, lower bound, upper bound)

Where:

expression is the expression to be minimized.
variable is the variable over which to minimize the expression.
lower bound is the lower bound of the interval over which to minimize the expression.
upper bound is the upper bound of the interval over which to minimize the expression.

For example, to find the minimum value of the function f(x) = x^2 over the interval [0, 1], you would use the following Minimum() function:

Minimum(x^2, x, 0, 1)

This function would return the value 0, which is the minimum value of f(x) over the interval [0, 1].

Using the Minimum() Function with Inequalities

The Minimum() function can also be used to find the minimum value of an expression subject to one or more inequalities. For example, to find the minimum value of the function f(x) = x^2 over the interval [0, 1] subject to the inequality x > 0.5, you would use the following Minimum() function:

Minimum(x^2, x, 0.5, 1)

This function would return the value 1, which is the minimum value of f(x) over the interval [0.5, 1].

Utilizing the Derivative to Locate Minimums

The derivative of a function can be used to find its minimums. A minimum occurs when the derivative is equal to zero and the second derivative is positive. To find the minimums of a function using the derivative:

Find the derivative of the function.
Set the derivative equal to zero and solve for x.
Evaluate the second derivative at the x-values found in step 2. If the second derivative is positive at that x-value, then the function has a minimum at that point.

For example, consider the function f(x) = x³ – 3x² + 2x.

The derivative of this function is f'(x) = 3x² – 6x + 2. Setting the derivative equal to zero and solving for x gives:

– 3x² – 6x + 2 = 0
– (3x – 2)(x – 1) = 0
– x = 2/3 or x = 1

Evaluating the second derivative f”(x) = 6x – 6 at these x-values gives:

x	f”(x)
2/3	0
1	6

Since the second derivative is positive at x = 1, the function has a minimum at x = 1. The minimum value is f(1) = 1.

Implementing the secant Method for Approximate Minimums

The secant method is an iterative method for finding the roots of a function. It can also be used to find the minimum of a function by finding the root of the function’s first derivative.

The secant method starts with two initial guesses for the root of the function, x1 and x2. It then iteratively improves these guesses by using the following formula:

““
x3 = x2 – f(x2) * (x2 – x1) / (f(x2) – f(x1))
““

where f(x) is the function being evaluated.

The method continues to iterate until the difference between x2 and x3 is less than some tolerance value.

The secant method is a relatively simple method to implement, and it can be very effective for finding the roots of functions that are differentiable. However, it can be sensitive to the choice of initial guesses, and it can fail to converge if the function is not differentiable.

Advantages of the secant method

Easy to implement
Can be very effective for finding the roots of functions that are differentiable

Disadvantages of the secant method

Can be sensitive to the choice of initial guesses
Can fail to converge if the function is not differentiable

Comparison of the secant method to other methods

The secant method is similar to the bisection method and the false position method. However, the secant method typically converges more quickly than the bisection method, and it is more robust than the false position method.

The following table compares the secant method to the bisection method and the false position method:

Method	Convergence rate	Robustness
Secant method	Quadratic	Good
Bisection method	Linear	Excellent
False position method	Quadratic	Poor

Employing Newton’s Method for Precise Minimums

Newton’s Method is a robust iterative process that converges rapidly to the minimum of a function. It utilizes the function’s first and second derivatives to refine approximations successively. The method starts with an initial guess and iteratively updates it based on the following formula:

x_n+1 = x_n – f(x_n) / f'(x_n)

where:

x_n is the current approximation
x_n+1 is the updated approximation
f(x) is the function being minimized
f'(x) is the first derivative of f(x)
f”(x) is the second derivative of f(x)

To use Newton’s Method in Desmos, follow these steps:

Define the function f(x) using the y= syntax.
Create a slider named “x” to represent the initial guess.
Define a function g(x) that represents the iterative formula:
```
g(x) = x - f(x)/f'(x)
```
Create a table that displays the iteration number, x_n, and the corresponding y-value f(x_n).
Animate the slider “x” by associating it with the input of g(x) and graphing the result.
As the animation progresses, the table will update with the iteration number and the corresponding minimum value.

Illustrative Example

Consider the function f(x) = x³ – 3x² + 2x + 1. Using Newton’s Method, we can find its minimum as follows:

Iteration	x_n	f(x_n)
0	1	1
1	0.6666666666666666	0.6666666666666666
2	0.4444444444444444	0.4444444444444444
3	0.2962962962962963	0.2962962962962963
…	…	…

As the number of iterations increases, the approximations converge rapidly to the minimum of f(x), which is approximately 0.296.

Leveraging the Optimization Palette

The Optimization Palette in Desmos is a powerful tool for finding the minimum or maximum values of functions. To use the Optimization Palette, simply click on the “Optimize” button in the toolbar, then select “Minimum”.

The Optimization Palette will then display a list of possible minimum values for the function. You can click on any of the values to see the corresponding x-value.

Here is a detailed breakdown of the steps involved in finding the minimum of a function using the Optimization Palette:

1. Enter the function into Desmos

The first step is to enter the function that you want to find the minimum of into Desmos. You can do this by clicking on the “>” button in the toolbar, then selecting “Function”.

2. Click on the “Optimize” button

Once you have entered the function, click on the “Optimize” button in the toolbar. This will open the Optimization Palette.

3. Select “Minimum”

In the Optimization Palette, select “Minimum”. This will tell Desmos to find the minimum value of the function.

4. Click on a value

The Optimization Palette will then display a list of possible minimum values for the function. You can click on any of the values to see the corresponding x-value.

5. (Optional) Change the domain

If you want to find the minimum of the function on a specific domain, you can change the domain in the Optimization Palette. To do this, click on the “Domain” button, then enter the new domain.

6. (Optional) Use advanced settings

The Optimization Palette also has a number of advanced settings that you can use to customize the optimization process. To access these settings, click on the “Advanced” button. The advanced settings include:

Setting	Description
Tolerance	The tolerance for the optimization process. A smaller tolerance will result in a more accurate solution, but will also take longer to compute.
Steps	The maximum number of steps that the optimization process will take. A larger number of steps will result in a more accurate solution, but will also take longer to compute.
Algorithm	The algorithm that the optimization process will use. There are two different algorithms available: the “Brent” algorithm and the “Golden Section” algorithm. The Brent algorithm is generally more efficient, but the Golden Section algorithm is more robust.

Identifying Multiple Minimums

To find multiple minimums in Desmos, you can use the following steps:

Graph the function.
Use the “Zoom” tool to zoom in on the area where you suspect there are multiple minimums.
Use the “Trace” tool to trace along the graph and find the minimum points.
The minimum points will be indicated by a small dot on the graph.
You can also use the “Table” tool to find the minimum points.
To do this, click on the “Table” icon and then click on the “Minimum” tab.
The table will show you a list of the minimum points and their corresponding x-values.

Here is an example of how to find multiple minimums in Desmos:

Steps	Image
Graph the function f(x) = x^2 – 4x + 3.
Use the “Zoom” tool to zoom in on the area where you suspect there are multiple minimums.
Use the “Trace” tool to trace along the graph and find the minimum points.
The minimum points are (1, -2) and (3, -2).

Customizing Minimum Output

If you only want the values of the minima of a function and not the x-coordinates, you can use the custom output option in the Function Analyzer tool. Here’s how:

Create a function in Desmos.
Click on the Function Analyzer tool in the top menu.
In the “Output” tab, select “Custom Output” from the dropdown menu.
Enter the following code in the “Custom Output” field:
“`
min(y)
“`
Click on the “Analyze” button.

The output will now show only the values of the minima of the function.

Example

Consider the function $f(x) = x^2 – 4x + 3$. To find the minimum of this function using custom output:

Enter the function in Desmos.
Open the Function Analyzer tool.
Select “Custom Output” in the “Output” tab.
Enter the code `min(y)` in the “Custom Output” field.
Click on the “Analyze” button.

The output will show the minimum value of the function, which is 1.

Using Table Output

Alternatively, you can use the table output option to get both the x-coordinates and the values of the minima. Here’s how:

Follow steps 1-2 from the previous method.
In the “Output” tab, select “Table” from the dropdown menu.
Set the “Table Interval” to a small value, such as 0.1.
Click on the “Analyze” button.

The output will now show the minima of the function in a table, including the x-coordinates and the values of the minima.

Finding X Minimums in Desmos

1. Introduction

Desmos is a free online graphing calculator that allows users to explore mathematics visually. One of the many features of Desmos is the ability to find the x-minimum of a function.

2. Finding the X Minimum of a Function

To find the x-minimum of a function in Desmos, follow these steps:

1. Enter the function into Desmos.
2. Click on the “Find Minimum” button.
3. Desmos will display the x-minimum of the function.

3. Applications of Finding X Minimums in Desmos

Applications of Finding X Minimums in Desmos

4. Finding the Minimum Value of a Function

The x-minimum of a function is the x-value at which the function has its minimum value. This can be useful for finding the minimum value of a function, such as the minimum cost of a product or the minimum time it takes to complete a task.

5. Finding the Turning Points of a Function

The x-minimum of a function is a turning point, where the function changes from decreasing to increasing. This can be useful for understanding the behavior of a function and for finding the maximum and minimum values of a function.

6. Finding the Roots of a Function

The x-minimum of a function is a root of the function, where the function has a value of 0. This can be useful for finding the solutions to equations and for understanding the zeros of a function.

7. Finding the Intercepts of a Function

The x-minimum of a function can be used to find the y-intercept of the function, which is the point where the function crosses the y-axis. This can be useful for understanding the behavior of a function and for finding the equation of a function.

8. Finding the Area Under a Curve

The x-minimum of a function can be used to find the area under the curve of the function. This can be useful for finding the volume of a solid or the work done by a force.

9. Optimization

Finding the x-minimum of a function can be used to optimize a function. This can be useful for finding the minimum cost of a product, the maximum profit of a business, or the minimum time it takes to complete a task.

Problem	Solution
Find the minimum value of the function f(x) = x^2 – 4x + 3.	The x-minimum of f(x) is x = 2, and the minimum value of f(x) is -1.
Find the turning points of the function g(x) = x^3 – 3x^2 + 2x + 1.	The x-minimum of g(x) is x = 1, and the x-maximum of g(x) is x = 2.
Find the roots of the function h(x) = x^2 – 5x + 6.	The x-minimum of h(x) is x = 2.5, and the roots of h(x) are x = 2 and x = 3.

Conclusion and Summary of Techniques

In conclusion, finding the x minimum in Desmos can be achieved using a variety of techniques. The most straightforward approach is to use the “minimum” function, which takes a list of values and returns the smallest one. However, this function can only be used to find the minimum of a single variable, and it cannot be used to find the minimum of a function. To find the minimum of a function, we can use the “solve” function. This function takes an equation and returns the value of the variable that satisfies the equation. We can use this function to find the minimum of a function by setting the derivative of the function equal to zero and solving for the value of the variable.

10. Finding the Minimum of a Multivariable Function

Finding the minimum of a multivariable function is a more complex task than finding the minimum of a single-variable function. However, it can be done using a similar approach. We can use the “solve” function to set the partial derivatives of the function equal to zero and solve for the values of the variables. Once we have found the values of the variables that satisfy the partial derivatives, we can plug these values back into the function to find the minimum.

Method	Description
Minimum function	Finds the minimum of a list of values.
Solve function	Finds the value of a variable that satisfies an equation.
Partial derivatives	The derivatives of a function with respect to each of its variables.

How To Find The X Minimum In Desmos

To find the x minimum of a function in Desmos, you can use the “minimum()” function. The syntax for the minimum() function is as follows:

minimum(expression, variable)