How do you determine if the function #sqrt (x(1-x))# defined on the interval [0,1] satisfies the hypotheses of the Mean Value Theorem, and then find the value or values that satisfy the equation #(f(b)-f(a))/(b-a)=f'(c)#?

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Answer 1 · 2015-10-09T02:29:25

Recall the hypotheses and answer the question.

Hypothesis 1: #f# is continuous on #[a,b}#

Is the function in this question contiuous on #[0,1]#. Why or why not?

Hypothesis 2: #f# is differentiable on #(a,b)#

Is the function in this question differentiable on #(0,1)#. Why or why not?

If these are true, you have some algwbra to do.

Find #f(0)# and #f(1)#, then find #(f(1)-f(0))/(1-0)#

Now find #f'(x)# (if you didn't already find it to prove that it exists)

Set #f'(x) = (f(1)-f(0))/(1-0)# and solve.

Any solution in the interval #(0,1)# is one of the #c#'s in the conclusion.