Mean Value Theorem Help!

Question

Verify the function f(x)=1/x satisfies the hypothesis of the mean value theorem on [1,3]. Then find all numbers c that satisfy the conclusion of the mean value theorem.

Answer 1

The mean value theorem states that the slope between the end points is the same as the derivative of the function at some point in between the end points of the function. 

 

Now evaluate f(1) = 1/1 =1, and f(3) = 1/3

 

The slope between the end points is (1/3 - 1)/(3 - 1) = -2/3 / 2 = -1/3

 

Now take the derivative of the function and equate it to the slope of the line that connects the end points in which is:

 

df(x)/dx=-1/x² =-1/3. This implies that x² = 3. Hence, x = sqrt(3). Therefore, at (sqrt(3),1/sqrt(3)) is the point in which satisfies the mean value theorem of the function f(x) = 1/x on [1,3].

Answer 2

f(x) = 1/x satisfies the conditions (hypothesis) to apply the mean value theorem: it's continuous and differentiable within [1, 3].

 

 

The mean value theorem says that there is some c between x = a and x = b such that

 

f'(c) = [f(b) - f(a)] / [b - a]

 

 

Let a = 1 and b = 3.

 

Then f(a) = f(1) = 1/1 = 1 and f(b) = f(3) = 1/3

 

 

[f(3) - f(1)] / [3 - 1] = [1/3 - 1] / 2 = -1/3.

 

 

Since f'(x) = -1/x²,

 

x² = 3   or   x = √3.