Given the function #f(x)=(-x^2+9)/(4x)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,3] and find the c?

2 Answers
Dec 30, 2016

The only hypothesis required by the mean value theorem is that #f(x)# has to be continuous in the interval.

Explanation:

The only hypothesis required by the mean value theorem is that #f(x)# has to be continuous in the interval.

As:

#f(x) = (9-x^2)/(4x)#

is in fact continuous in #[1,3]# the hypothesis is satisfied and we have that the is a value #c in [1,3]# for which:

#f(c) = 1/2 int_1^3 (9-x^2)/(4x)dx#

We can calculate the definite integral as:

#int_1^3 (9-x^2)/(4x)dx = int_1^3 (9/(4x)-x/4)dx=[9/4lnx-x^2/8]_1^3= 9/4ln3-9/8+1/8=9/4ln3-1#

Then we can find #c# from:

#f(c) = (9-c^2)/(4c) = 9/4ln3-1#

Dec 31, 2016

I will assume that you are referring to the Mean Value Theorem for Derivatives.

Explanation:

You determine whether it satisfies the hypotheses by determining whether #f(x) = (-x^2+9)/(4x)# is continuous on the interval #[1,3]# and differentiable on the interval #(1,3)#. (Those are the hypotheses of the Mean Value Theorem)

You find the #c# mentioned in the conclusion of the theorem by solving #f'(x) = (f(3)-f(1))/(3-1)# on the interval #(1,3)#.

#f# is continuous on its domain, which includes #[1,3]#

#f'(x) = -(x^2+9)/(4x^2)# which exists for all #x != 0# so it exists for all #x# in #(1,3)#

Therefore this function satisfies the hypotheses of the Mean Value Theorem on this interval.

To find #c# solve the equation #f'(x) = (f(3)-f(1))/(3-1)#. Discard any solutions outside #(1,3)#.

You should get #c = sqrt3#. (The solution, #-sqrt3# is not in the interval.)