What are the critical values for the function fx 2x3x2 x1x

What are the critical values for the function f(x) = (2x-3)/(x-2) + (x+1)/(x+2)

Given f(x)= (2x-3)/(x-2) + (x+1)/(x+2)

First we will rewrite with a common denominator (x-2)(x+2) = x^2-4

==> f(x)= (2x-3)(x+2) + (x+1)(x-2)]/ (x^2-4)

==> f(x) = ( 2x^2 +x -6 + x^2 -x -2) / (x^2-4)

==> f(x)= ( 3x^2 -8)/(x^2-4)

Now we will calculate the first derivative.

==> f\'(x)= ( 6x)(x^2-4) - (3x^2-8)(2x)/ (x^2-4)^2

==> f\'(x)= ( 6x^3 - 24x - 6x^3 + 16x) / (x^2-4)

==> f\'(x)= ( -8x)/(x^2-4)

Now we will find the zeros.

==> -8x = 0 ==> x= 0

Then the critical values of the function is x= 0.