Why is a New Function Created in the Proof of the Mean Value Theorem?

[Bashyboy]

I am reading the proof for the M.V.T, mostly understanding it all, except for this one step. Here is the link to it: http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeAppsProofs.aspx#Extras_DerAppPf_MVT
It's near the bottom of the page.

What I don't precisely is why they create a new function, g(x), which is defined as f(x) subtracted by the equation for the secant line. A few steps after this they are able to redefine the interval (a, b) for this new function g(x), where the endpoints are equal to equal to each other, but I just don't understand the motive for this. What does it accomplish in proving this theorem?

 

[Millennial]

The function is necessary if Rolle's Theorem will be applied.

 

[Infinitum]

Expanding on what Millenial is saying, g(x) was defined so that you could directly use the result of Rolle's Theorem to help prove the mean value theorem. This is assuming you already know Rolle's theorem, whose proof is given above. Basically, if you do have g(x), you cannot use the fact that Rolle's theorem is applicable(it is only applicable on functions, after all!) and hence be unable to deduce a proof MVT.

 

[Bashyboy]

Okay, so think in terms of Rolle's Theorem when lighting upon that step. Alright, I understand. Thank you both.