How do you find the values of C guaranteed by the Mean Value Theorem for #f(x)= 9/x^3# over [1, 3]?

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Answer 1 · 2015-09-12T10:54:19

In the conclusion of the theorem, #c# is a solution in #(1,3)# to the equation #f'(x) = (f(3)-f(3))/(3-1)#

So write the equation and solve it.

#-27/x^4 = (1/3 - 9)/(3-1) = (-26/3)/2 = -13/3#

So

#13x^4 = 81# and

#x = +- 3/root(4)13#

Obviously, the negative solution is not in the interval, so there is only one value of #c#, namely

#c=3/root(4)13#

You should convince yourself that

#1 < 3/root(4)13 < 3#