Question

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?

Answer 1

`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)`.