Calculating the first and second derivatives and building the variation charts
#y=(x^2+1)/(x^2-4)#
The domain of #y# is #x in {RR -(-2,2)}#
#u(x)=x^2+1#, #=>#, #u'(x)=2x#
#v(x)=x^2-4#, #=>#, #v'(x)=2x#
Therefore,
#dy/dx=(u'v-uv')/(v^2)=(2x(x^2-4)-2x(x^2+1))/(x^2-4)^2#
#=(2x^3-8x-2x^3-2x)/(x^2-4)^2#
#=(-10x)/(x^2-4)^2#
The critical points are when #dy/dx=0#
#=>#, #-10x=0#, #x=0#
Building the variation chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaaaa)##-2##color(white)(aaaaaaaa)##0##color(white)(aaaaaaaaa)##2##color(white)(aaaaaaa)##+oo#
#color(white)(aaaa)##dy/dx##color(white)(aaaaaaa)##+##color(white)(aaaa)##||##color(white)(aaa)##+##color(white)(aaa)##0##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aaaa)##-#
#color(white)(aaaa)##y##color(white)(aaaaaaaaa)##↗##color(white)(aaa)##||##color(white)(aaa)##↗##color(white)(a)##-1/4##color(white)(aaaa)##↘##color(white)(aaa)##||##color(white)(aaaa)##↘#
Calculating the second derivative to determine the concavity and the points of inflections
#u(x)=-10x#, #=>#, #u'(x)=-10#
#v(x)=(x^2-4)^2#, #=>#, #v'(x)=4x(x^2-4)#
#(d^2y)/(dx^2)=(u'v-uv')/(v^2)=(-10(x^2-4)^2+40x^2(x^2-4))/(x^2-4)^4#
#=((x^2-4)(-10x^2+40+40x^2))/(x^2-4)^4#
#=(10(3x^2+4))/(x^2-4)^3#
When #x=0#, #=>#, #(d^2y)/(dx^2)<0#, this is a local max
Building the variation chart
#color(white)(aaaa)##Interval##color(white)(aaaa)##(-oo,-2)##color(white)(aaaa)##(-2,2)##color(white)(aaaa)##(2,+oo)#
#color(white)(aaaa)##Sign (d^2y)/(dx^2)##color(white)(aaaaaaaa)##+##color(white)(aaaaaaaaaa)##-##color(white)(aaaaaaaa)##+#
#color(white)(aaaa)##y##color(white)(aaaaaaaaaaaaaaa)##uu##color(white)(aaaaaaaaaa)##nn##color(white)(aaaaaaaa)##uu#
See the graph below
graph{(x^2+1)/(x^2-4) [-7.9, 7.9, -3.95, 3.95]}
