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

1 Answer
Dec 4, 2016

You determine whether it satisfies the hypotheses by determining whether #f(x) = 9/x^3# is continuous on the interval #[1,3]# and differentiable on the interval #(1,3)#.

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) = (-27)/x^4# 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)#.

I believe that you should get #c = root(4)(54/13)#.