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.

 

[blue_raver22]

I can understand your confusion about this step in the proof for the Mean Value Theorem. Let me explain the rationale behind creating a new function, g(x), and how it helps in proving the theorem.

Firstly, it is important to understand that the Mean Value Theorem states that for any continuous and differentiable function, there exists a point within the interval (a, b) where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints. In other words, there exists a point c within the interval (a, b) where the derivative of the function, f'(c), is equal to the average rate of change of the function, [f(b)-f(a)]/(b-a).

Now, in order to prove this theorem, we need to show that there exists a point c where the derivative of the function is equal to the average rate of change. This is where the new function, g(x), comes into play. By defining g(x) as f(x) subtracted by the equation for the secant line, we are essentially creating a new function that represents the difference between the actual function and the average rate of change.

Next, when we redefine the interval (a, b) for this new function g(x), we are essentially looking at a smaller interval within the original interval (a, b). This smaller interval will have the same endpoints as the original interval, but the function values within this interval will be different due to the subtraction of the secant line equation. This allows us to focus on a smaller portion of the function and analyze it more closely.

Now, if we can show that there exists a point c within this new interval where the derivative of g(x) is equal to 0, then we have essentially proved the Mean Value Theorem. This is because if the derivative of g(x) is equal to 0 at some point c, then the function g(x) is flat at that point, which means that the difference between the actual function and the average rate of change is 0 at that point. And since the endpoints of the interval are equal to each other, this means that the average rate of change is also 0, which is exactly what the Mean Value Theorem states.

In conclusion, creating the new function g(x) and redefining the interval (a, b) allows us to focus on a smaller portion of the function and analyze it more closely