Ap Calc help

[zxcvbnm1216]

I am supposed to do a curve sketch of the following function:

f(x)= (-5x^2+3x)/(2x^2-5)

We have to show all our work including asymptotes, intercepts, derivatives, periods of increasing and decreasing, max/min values, critical points, points of inflection and tangent slope points.

I got the x intercept:
x=0
Y intercept:
y=0
VA:
at x= root(2.5)
HA:
at y= -2.5
No oblique/slant asymptote
First derivative:
-(6x^2-50x+15)/((2x-5)^2)

I have to find critical points by factoring the first derivative. This would show me the period of increasing and decreasing and max/min values but I do not know how to do this because isn’t this derivative un-factorable? What should I do?

[zxcvbnm1216]

@MITer94

[MITer94]

This would show me the period of increasing and decreasing and max/min values but I do not know how to do this because isn’t this derivative un-factorable? What should I do?

@zxcvbnm1216 I hope you remember how to solve quadratic equations. Same thing applies to some other parts - for example, there is more than one x-intercept.

Also remember that f’(x) equaling zero is not the only condition for finding max/min values. You have to show that f is actually defined at that critical point, and that f actually contains a local minimum/maximum at that point (usually occurs when f’ changes sign in the neighborhood of x).