Question

What are the critical points of `(x^2/(x^2-1))`?

Answer 1

I will put f(x) in front of `(x^2/(x^2-1))`

To find the critical number, you must get the first derivative of
f(x) = `(x^2/(x^2-1))`

The first derivative is `f^'(x) = (-2x)/(x^2-1)^2`

Now you must set `f^'(x) = 0`, and you must also find where the `f^'(x)` does not exist (dne).

`f^'(x) = 0` ------------------------- `f^'(x)` dne

`(-2x)/(x^2-1)^2` = 0 ------------------`f^'(x)` dne at x = 1 and x = -1

-2x = 0
x = 0 ------------------------------now you must plug them back to the
---------------------------------------- back to the original equation.
---------------------------------------Since x = 1 and x = -1 are undefined, they --------------------------------------------------------are not critical numbers

Since x = 0 is defined in the original equation, it is a critical number.