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

Question

Ezra Adams · Accepted Answer

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

The domain of the function is the entire #RR#, as the denominator of the rational function is always #>0#. We have that: #lim_(x->-oo) x/(x^2+1) = 0# #lim_(x->+oo) x/(x^2+1) = 0# We can see that #y(x)# has the line #y=0# as horizontal asymptote on both sides. We can also see that: #y(x) < 0 # for #x<0# #y(x) > 0 # for #x>0# #y(x) = 0 # for #x=0# So #x=0# is the only intercept. #y'(x) = frac(1-x^2) ((x^2+1)^2)# As the denominator of #y'(x)# is always positive the function is differentiable everywhere and: #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 #y(x)# starts from #y=0# at #x->-oo# and decreases until #x=-1# where it reaches a local minimum #y(-1)=-1/2#. It then increases until #x=1# (changing sign at #x=0#) where it reaches a local maximum at #y(1) = 1/2#, and then decreases approaching zero indefinitely as #x->+oo# #y''(x) = frac (2x(x^2-3)) ((x^2+1)^3)# so inflection points are: #x=0# and #x=+-sqrt(3)# and the concavity is determined by the sign of #y''(x)#: 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

Axel Alvarez · Answer

undefined To find the intercepts, set $ y = 0 $ and solve for $ x $. For extrema, differentiate the function, set the derivative equal to zero, and solve for $ x $. To find points of inflection, differentiate the function twice, set the second derivative equal to zero, and solve for $ x $. To find asymptotes, determine the horizontal asymptote by analyzing the behavior of the function as $ x $ approaches positive or negative infinity, and vertical asymptotes by finding values of $ x $ that make the denominator of the function equal to zero. To graph $ y = \frac{x}{x^2 + 1} $, plot intercepts, extrema, points of inflection, asymptotes, and sketch the curve accordingly.