1. The problem statement, all variables and given/known data Find the absolute max/min of the function on it's interval. [tex]f(x)=\frac{x^2-4}{x^2+4}[/tex] Interval: [-4,4] 2. Relevant equations 3. The attempt at a solution [tex]f(x)=\frac{x^2-4}{x^2+4}[/tex] I basically want to find all the critical points, so I set the denominator to zero and found a critical point to be where x = 2i, and x = -2i. Then I took the derivative of the function as so: [tex]f'(x) = \frac{2x(x^2+4)-2x(x^2-4)}{(x^2+4)^2}[/tex] Setting the numerator to zero should find where the derivative is equal to zero, but that expands out to this: [tex]2x^3+8x-2x^3+8x[/tex] Which is basically zero anyway. I was going to try to factor out something from the numerator and denominator (if possible) to cancel it out so I could get something, but I thought I would lose solutions doing that. So here is where I am confused? I tried graphing it, thinking that it would just return the line y=0, but my TI-89 just says "not a function or program, error"? edit: fixed
Your algebra is faulty. The x terms don't cancel. And you probably don't care about imaginary solution in the denominator.
[tex]2x^3+8x-2x^3+8x=0[/tex] from the numerator, works out to be: [tex]16x=0[/tex] x=0 I was not sure what to do with the imaginary solutions, essentially the only critical point found is at x=0 then I would imagine. Leaving me to check the endpoints and x=0 so find the absolutes correct?
Alright, I think I got it now. This was the only one in the set of problems that had the possibility of an imaginary solution, and I really wasn't sure if that counted or not. Thanks for the help.