Question

Problem 55b

Answer 1

Given information

We have

 

  

 

F and G be any two antiderivatives of f on I

 

Step-by-step explanation

Step 1.

We have

 

  

 

F and G be any two antiderivatives of f on I

  

 

So...

 

  

    

 

In  interval I according to Lagrange´s mean value theorem we have atleast one point α

 

  

 

We have from (1)

 

  

 

So we have

 

  

 

From (1) we have

 

  

 

Hence antiderivative of 0 being H(x), we get

 

 

Hence we have

 

  

 

 

Since G(x) is an antiderivative of f(x) we have proved Theorem 1 from above.