Please state the following components of this rational function?
-intervals of increase/decrease
-local maximum and minimum values
-intervals of concave up/concave down
-points of inflection
The Attempt at a Solution
There is a horizontal asymptote at y=6.
There are vertical asymptotes at x=-3 and x=5
The first derivative is -12x(x+15)/(x^2-2x-15)^2
The second derivative is 12(2x^3+45x^2+225)/(x^2-2x-15)^3
According to the first derivative, the critical numbers would be -15, and 0.
After the first derivative test, I found out that a min. point occurs when x=-15 and a max. point occurs when x=0. However, I got (-15,5.625) and (0,0). How can the min. point have a larger y-value than the max. point?
Also, when I graphed this function on GraphCalc, it seems that there does not seem to be a minimum point at x=-15. Why is that?
When I graphed the function, there seems to be 3 branches; in the left, middle, and right. The left branch seems to go through the horizontal asymptote (y=6) but then it slowly approaches it from below y=6.
Please graph f(x)=6x^2/(x^2-2x-15) at http://math.ucalgary.ca/undergraduate/webwork/graphing-calculator [Broken] to see what I am describing. I cannot justify why there would be a minimum at x=-15.
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