How do you find intercepts, extrema, points of inflections, asymptotes and graph #g(x)=x+32/x^2#?

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Answer 1 · 2017-12-01T17:23:57

Please see below for a partial solution.

#g(x) = x+32/x^2#

x intercepts
#g(x) = 0# at #x=-32/x^2#
which happens at #x^3 = -32#

so #x = root(3)(-32) = -2root(3)4#

y intercept
None. #g(0)# does not exist.

Asymptotes

#lim_(xrarr0)g(x) = oo#, so #x=0# (the #y#-axis) is a verticle asymptote..

#lim_(xrarr00)g(x) = oo# so there is no horizontal asymptote.

#lim_(xrarroo)(g(x)-x) = 0# so #y=x# is an oblique (slant) asymptote)

Analysis of first derivative

#g'(x) = 1-64/x^3 = (x^3-64)/x^3# is undefined at #x=0# and is #0# at #x=4#.

On #(-oo,0)#, we have #g'(x) > 0# so #g# is increasing.
#x=0# is not a critical number.

On #(0,4)#, we have #g'(x) < 0# so #g# is decreasing.
On #(4,oo)#, we have #g'(x) > 0# so #g# is increasing.

#f(4)=6# is a local minimum.