# How to maximize a multivariable equation?

• DyslexicHobo
In summary, the conversation discusses finding the maximum value of a function with 4 variables, which is represented by a quotient of two functions. The person has access to MathCAD and Mathematica software but is not familiar with using Mathematica. The process of finding critical points for multivariable functions is similar to that of single variable functions, where the derivative is set to zero. In multivariable functions, this involves solving a system of equations. The example of a bowl is used to illustrate the concept of critical points in multivariable functions.
DyslexicHobo
I have an function with 4 variables. Each of the 5 variables are bounded between to real numbers. Is there an easy way to find a maximum value of the function?

Here is the equation: http://yfrog.com/jleqnp

I have MathCAD and Mathematica software available to me. I'm not very familliar with how to use Mathematica, though.

Thanks for any help.

Edit: I just realized that the function shown is actually the quotient of two functions. The left side is what I need to maximize (I(x1,x2,x3,x4) / A(x1,x2,x3,x4)). It can be treated as one function. I just didn't define it as so.

Finding critical points (min/max/inflection) of multivariable functions is an extension of single variable functions. For single variable functions, critical points are where the derivative of the function with respect to the variable is equal to zero.

For multivariable functions, critical points are where the partial derivatives of the function with respect all independent variable is equal to zero. You thus end up with a system of equations to solve.

As a simple example, think of a bowl. Pick an arbitrary point on the bowl and imagine two planar slices a right angles through the bowl. The profile of each slice will look sort of like a parabola but the location of the minimum point in each slice will be different. Only when the slices go through the bottom of the bowl will the minimum points of the two slices be coincident.

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## 1. How do I identify the independent and dependent variables in a multivariable equation?

The independent variables are the ones that you can change or manipulate in the equation, while the dependent variable is the one that is affected by the changes in the independent variables.

## 2. What is the best way to simplify a multivariable equation?

The best way to simplify a multivariable equation is to first identify any common factors and then combine like terms. You can also use algebraic rules and properties to rearrange the equation and eliminate any unnecessary terms.

## 3. How can I determine the maximum value of a multivariable equation?

To determine the maximum value of a multivariable equation, you can use calculus techniques such as taking the derivative and setting it equal to zero to find the critical points. Then, plug these critical points back into the original equation to determine the maximum value.

## 4. Is it possible to maximize a multivariable equation without using calculus?

Yes, it is possible to maximize a multivariable equation without using calculus. You can use graphical methods such as plotting the equation and finding the highest point on the graph, or you can use algebraic methods such as completing the square or using the quadratic formula.

## 5. What are some real-world applications of maximizing multivariable equations?

Maximizing multivariable equations is used in various fields such as economics, engineering, and physics. It can be used to optimize production processes, determine the most profitable business strategies, and find the maximum efficiency in a system. It is also commonly used in mathematical modeling and data analysis.

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