# How do you find intercepts, extrema, points of inflections, asymptotes and graph #y=x/(x^2+1)#?

##### 1 Answer

graph{x/(x^2+1) [-10, 10, -1, 1]}

#### Explanation:

The domain of the function is the entire

We have that:

We can see that

#y(x) < 0 # for#x<0#

#y(x) > 0 # for#x>0#

#y(x) = 0 # for#x=0#

So

As the denominator of

#y'(x) <0# for#x in (-oo,-1)# and#in in (1,+oo)#

#y'(x) >0# for#x in (-1,1)#

#y'(x) = 0# for#x=+-1#

Therefore

so inflection points are:

for

#x in (-oo, -sqrt(3)), y''(x) <0, y(x)# is concave down

for#x in (-sqrt(3),0), y''(x) >0, y(x)# is concave up

for#x in (0, sqrt(3)), y''(x) <0, y(x)# is concave down

for#x in (sqrt(3),+oo), y''(x) >0, y(x)# is concave up