# Proof max{f(x),g(x)}=1/2[(f + g) + |f - g|]

• QUBStudent
In summary, the notation "max{f(x),g(x)}" means to take the larger value between f(x) and g(x). To prove the equation max{f(x),g(x)}=1/2[(f + g) + |f - g|], we can use the definition of max function and rearrange the equation. The absolute value is significant in ensuring a positive value for the difference between f(x) and g(x). This equation can be applied to functions with multiple variables and is useful in real-world applications such as optimization and statistics.
QUBStudent
hi, max{f(x),g(x)}=1/2[(f + g) + |f - g|] is the equation of the maximum of two functions on the real axis. Can anyone give me a hint on how to show where this equation comes from or how it is derived

Let M(x) be your max. function.
Suppose that, for a particular choice of x, f(x)>=g(x).
Then, |f-g|=f-g, from which follows that M(x)=f(x).
If g(x)>=f(x), then |f-g|=g-f, that is, M(x)=g(x)

cool thnx for the response

## 1. What does the notation "max{f(x),g(x)}" mean?

The notation "max{f(x),g(x)}" means to take the larger value between f(x) and g(x). It is used to find the maximum value between two functions.

## 2. How do you prove max{f(x),g(x)}=1/2[(f + g) + |f - g|]?

To prove this equation, we can use the definition of max function: max{a,b} = 1/2(a+b+|a-b|). By substituting f(x) and g(x) for a and b, we get max{f(x),g(x)} = 1/2(f(x)+g(x)+|f(x)-g(x)|). Since max{f(x),g(x)} is equal to the larger value between f(x) and g(x), we can rearrange the equation to get max{f(x),g(x)} = 1/2[(f+g) + |f-g|].

## 3. What is the significance of the absolute value in the equation?

The absolute value ensures that we always get a positive value for the difference between f(x) and g(x). This is important because we want to find the larger value between the two functions, regardless of whether it is f(x) or g(x).

## 4. Can this equation be applied to functions with multiple variables?

Yes, this equation can be applied to functions with multiple variables as long as the functions are continuous and differentiable. The max function can be extended to multiple variables by taking the maximum value of each variable.

## 5. How is this equation useful in real-world applications?

This equation is useful in real-world applications where we need to compare two functions and determine the maximum value between them. For example, in optimization problems, we may need to find the maximum value of a function to determine the best solution. This equation can also be used in statistics to compare two sets of data and find the larger value between them.

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