# Thread: Don't understand use of auxiliary function in proving theorems

1. ## Don't understand use of auxiliary function in proving theorems

How was the auxiliary function to be used in proving the theorem below determined? I don't understand why this function was chosen, and how the author arrived at the decision of choosing this function? This is the first time I came across the use of auxiliary functions in proving a theorem, so I am a bit confused whether the function is chosen by pure intuition or is there a method behind this.

... so I am a bit confused whether the function is chosen by pure intuition or is there a method behind this.
The important thing is that $\displaystyle F$ works and you understand why works. All the same, in this case, analyze:

1.- $\displaystyle G(x)=f(x)-f(a)$ translates vertically the graph of $\displaystyle f$ in such a way that $\displaystyle G(a)=0$ .

2.- The equation of the straight line containing the endpoints of the $\displaystyle G$ graph is:

$\displaystyle y=\dfrac{f(b)-f(a)}{b-a}(x-a)$

Regards.

Fernando Revilla

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### what is an auxiliary function in mathematics

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