Question

What are the critical values, if any, of `f(x)=(x^2-3x+6)/(x-2)`?

Answer 1

`0` and `4`

Explanation:

The critical values of a function are numbers in the domain of the function at which the derivative does not exist or the derivative is `0`.

The domain of `f(x)=(x^2-3x+6)/(x-2)` is `RR - {2}`.

`f'(x) = ((2x-3)(x-2)-(x^2-3x+6)(1))/(x-2)^2`

`= (x^2-4x)/(x-2)^2`

`f'` fails to exist at `2` which is not in the domain of `f`. and

`f'(x) = 0` at `0` and `4`.

So, the critical value for `f` are `0` and `4`.