Given the function #f(x) = 1 / (x-1)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [2,5] and find the c?

1 Answer
Ratnaker Mehta
Aug 13, 2016

#c=3 in (2,5)#.

Explanation:

The Mean Value Theorem states that, : If a function #f : [a,b] rarr RR# is (1) continuous on #[a,b]#, (2) derivable on #(a,b)#, then there exists at least one #c in (a,b)# such that,

#(f(b)-f(a))/(b-a)=f'(c)#

We see that, the given fun. #f# is a rational fun., and hence, it is cont. on [2,5] and differentiable on #(2,5)#.

Therefore, by MVT, there must exist a #c in (2,5)#, s.t.#f'(c)=(f(5)-f(2))/(5-2)={1/(5-1)-1/(2-1)}/(3)=1/3(1/4-1)=1/3(-3/4)=-1/4..........(1)#

And, #f(x)=1/(x-1) rArr f'(x)=-1/(x-1)^2........................(2)#

Hence, by #(1) and (2)#, #-1/(c-1)^2=-1/4 rArr (c-1)^2=4#

#rArr c-1=+-2 rArr c=1+-2 rArr c=3, or, -1#

Since, #c=-1 !in (2,5),# we get, #c=3 in (2,5)#.

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