True or false question

Question

Rolle's Theorem does not apply for f (x) = 4/x-3, in the interval [1, 5]

Answer 1

f(x) = 4/(x-3) in the interval [1,5]

f(1) =4/-2 = -2

f(5) = 4/(5-3) = 4/2 = 2

f(1) does not equal f(5) as required by Rolle's Theorem

also f(x) is discontinuous at x=3, in the given interval

Rolles' Theorem requires the fuction be continuous over the closed interval

Rolles' Theorem doubly does not apply

3rd requirement is that f(x) be differentiable over the open interval (1,5)

triple problems

UNLESS you really did mean f(x) = 4/x -3 as (4/x) -3, not 4(x-3)

then f(5) = 4/5 -3 = -2.2

and f(1) = 1/5-3 = -.2.8

closer but Rolle's Theorem still doesn't apply

Answer 2

Hello,

Rolle's theorem, which is a special case of the more general mean value theorem, requires that the function be continuous within the interval of interest. the function f(x) = 4/x-3 is not continuous when the denomenator is 0 or when x = 3. x = 3 belongs to the interval [1,5] so the function is not continuous and as such Rolle's theorem cannot be applied. Thus, the statement is true.

I hope this helps!

Best,

Ibrahim