Mean value theorem

Question

What is the values of c using the mean value theorem.  F(x)=-x+(1/x) on 1/2less than or equal to x less than equal to 4 

Answer 1

f(x) = -x + 1/x is continuous on [1/2, 4] and is differentiable on (1/2, 4).  So, the Mean Value Theorem applies.  

 

Therefore, there exists at least one number, c, in the interval (1/2, 4) such that f'(c) = [f(4) - f(1/2)]/[[4-1/2]

 

So, -1 - 1/c² = [-3.75 -1.5]/3.5

 

      -1 - 1/c² = -3/2

 

          -1/c² = -1/2

 

              c² = 2

 

            c = ±√2

 

-√2 is not in the interval under consideration, so c = √2.