If #( x+2) / x#, what are the points of inflection, concavity and critical points?

1 Answer

Please see the explanation below

Explanation:

Let #f(x)=(x+2)/x#

The domain of #f(x)# is #x in RR-{0}#

The critical points are when #f'(x)=0#

#f'(x)=(1*x-(x+2)+1)/(x^2)=-2/x^2#

#f'(x)!=0#

There are no critical points

The points of inflection are when #f''(x)=0#

#f''(x)=4/x^3#

#f''(x)!=0#

There are no points of inflections

The concavity will depend on the sign of #f''(x)#

Let's build a sign chart

#color(white)(aaaa)##"Interval"##color(white)(aaaa)##(-oo,0)##color(white)(aaaa)##(0, +oo)#

#color(white)(aaaa)##"Sign " f''(x)"##color(white)(aaaa)##-##color(white)(aaaaaaa)##+#

#color(white)(aaaa)##"Concavity"##color(white)(aaa)##concave##color(white)(aaaa)##convex#

graph{(x+2)/x [-10, 10, -5, 5]}