How prove the maximum of 2 functions

In summary, the maximum of two functions can be proven by comparing their derivative functions and finding their critical points. The larger derivative function at the critical points will determine the maximum of the original functions. This method can be applied to any type of function, whether it is linear, quadratic, or exponential. Additionally, graphical representations can also be used to visually determine the maximum point of two functions. By finding the intersection point of the two graphs, the maximum value can be identified.
  • #1
lmamaths
6
0
Hi,

The maximum of 2 functions, namely f(x) and g(x) is defined as…

M(x) = ½ [ ( f + g ) + | f – g | ]

Can someone show me how this is proved out
or derived please?

Many Thx! :smile:
 
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  • #2
(a) Can you prove it for numbers?
I.E. max(x, y) = ((x + y) + |x - y|) / 2

(b) Once you've done that, can you then prove it for functions?
 
  • #3
How to prove the maximum of 2 functions

Thx Hurkyl,

Yes you are correct it works for both numbers as well as functions. But what I need to know is where or how did this equation come about – that is, do you know how it was created from First Principles – and please tell me how this is done if you know!

Many thx!
 

Related to How prove the maximum of 2 functions

1. How do you determine the maximum of two functions?

In order to determine the maximum of two functions, you need to first find the critical points of each function. These are the points where the derivative of the function equals zero or does not exist. Then, you can compare the values of the functions at the critical points and at the endpoints of the interval to determine which function has the higher maximum value.

2. What is the importance of finding the maximum of two functions?

Finding the maximum of two functions is important in many real-world applications, such as optimization problems in engineering and economics. It also allows us to compare the performance or effectiveness of two different functions and make informed decisions.

3. Can the maximum of two functions be the same?

Yes, it is possible for two functions to have the same maximum value. This can occur when the functions have the same critical points and their values at those points are equal. It can also occur at the endpoints of the interval if both functions have the same value at those points.

4. Is there a specific method for proving the maximum of two functions?

Yes, there are various methods for proving the maximum of two functions, such as using calculus techniques like finding critical points, graphing the functions to visually compare their maximum points, or using algebraic techniques like finding the roots of the difference between the two functions.

5. Can the maximum of two functions change for different intervals?

Yes, the maximum of two functions can change for different intervals. This is because the critical points and endpoints of the interval may be different, resulting in different maximum values for the functions. It is important to consider the specific interval when trying to prove the maximum of two functions.

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