Question

How do you verify the intermediate value theorem over the interval [5/2,4], and find the c that is guaranteed by the theorem such that f(c)=6 where `f(x)=(x^2+x)/(x-1)`?

Answer 1

Please see the verification in the explanation section.

Explanation:

`f` is continuous on `[5/2,4]`

Proof : `f` is a rational function and rational functions are continuous on their domains. `f` is continuous at all reals except `1`. (Its domain is all reals except `1`.) Since `1` is not is `[5/2,4]`, `f` is continuous on that closed interval.

`6` is between `f(5/2)` and #f(4)

Proof: `f(5/2) = 35/6 < 36/6 = 6` and `f(4) = 20/3 > 18/3 = 6`

Therefore, there is a `c` in `(5/2,4)` with `f(c) = 6`

To find the `c` (or `c`'s), solve the equation:

`f(x) = 6`.

Discard values outside `(5/2,4)`.

Note the solutions are `2` and `3`, but only `3` is in the interval.