Given the function #f(x)=abs(x-3)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [0,4] and find the c?

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Answer 1 · 2017-04-12T12:22:01

#f(x)# does not satisfy the hypotheses of the Mean Value Theorem.

The #MVT# requires that the function #f(x) = abs(x-3)# is continuous on the interval #[0,4]#, and differentiable in the open interval #(0,4)#

In this case, #f(x)# is continuous in the interval, but is not differentiable for #x=3#, so the #MVT# hypotheses are not satisfied.

In fact:

#(f(4)-f(0))/(4-0) = (1-3)/4 = -1/2#

But:

#f'(x) = {(-1 " for " 0 <= x < 3),(1 " for " 3 < x <= 4):}#

and nowhere in the interval we have #f'(x) = -1/2#