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

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Answer 1 · 2016-11-11T20:44:43

See below.

#f# is continuous at every real number except #0#.
#0# is not in #[1,4]#.
Therefore, #f# is continuous on #[1,4]#

#f'(x) = 8x^-3# is defined for all #x# other than #0#.
#0# is not in #[1,4]#.
Therefore, #f# is differentiable on #(1,4)#

To find #c#, solve #f'(x) = (f(4)-f(1))/(4-1)#. The solutions in #(1,4)# are values of #c#.

Solving, we get #x=root(3)(32/5)#