Given the function #f(x)=x^(2/3)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [0,1] and find the c?

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Answer 1 · 2017-01-06T20:33:10

#f# is the composition of functions that are continuus on #RR#, so it is continuous on #[0,1]#.

#f'(x)# exists for all #x# other than #0#, so #f# is differentiable on #(0,1)#

Therefore, #f# satisfies the hypotheses of the Mean Value theorem.

Therefore there is a #c# in #(01,)# such that #f'(c) = (f(1)-f(0))/(1-0)#

To actually find the #c#, solve the equation on the interval.

You should get #c=8/27#