We need
(1/x)'=-1/x^2
The domain of y is D_y=RR-{0}
We calculate the first derivative
y=x+4/x
dy/dx=1-4/x^2
To find the critical points, we calculate the values of x when dy/dx=0
when
1-4/x^2=0
1=4/x^2
x^2=4
Therefore, x=-2 and x=2
We can build the chart
color(white)(aaaa)xcolor(white)(aaaa)-oocolor(white)(aaaa)-2color(white)(aaaaaaaa)0color(white)(aaaaaaa)2color(white)(aaaaa)+oo
color(white)(aaaa)x+2color(white)(aaaaaa)-color(white)(aaaa)+color(white)(aaaa)||color(white)(aaa)+color(white)(aaaa)+
color(white)(aaaa)x-2color(white)(aaaaaa)-color(white)(aaaa)-color(white)(aaaa)||color(white)(aaa)-color(white)(aaaa)+
color(white)(aaaa)dy/dxcolor(white)(aaaaaaaa)+color(white)(aaaa)-color(white)(aaaa)||color(white)(aaa)-color(white)(aaaa)+
color(white)(aaaa)ycolor(white)(aaaaaaaaaa)↗color(white)(aaaa)↘color(white)(aaa)||color(white)(aaa)↘color(white)(aaaa)↗
The intervals of increasing are x in ]-oo,-2[uu]2,+oo[
The intervals of decreasing are x in ]-2,0[uu]0,2[
