How do you sketch the curve #y=x+1/x# by finding local maximum, minimum, inflection points, asymptotes, and intercepts?

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Nov 5, 2016

Answer:

Please see the explanation.

Explanation:

Here is a graph of the curves:

graph{x+1/x}

The first derivative is:

#dy/dx = 1 - 1/x^2#

To obtain the x coordinates of local extrema, set that equal to zero:

#1 -1/x^2 = 0#

#1 = 1/x^2#

#x^2 = 1#

#x = +-1#

#y(-1) = -1 + 1/-1 = -2#

#y(1) = 1 + 1/1 = 2#

Perform the second derivative test:

#(d^2y)/(dx^2) = 1/x^3#

It is positive for #x = 1# and negative for #x = -1#

This makes the point #(1,2)# a local minimum and the point #(-1,2)# a local maximum. Within domain #-1 <= x <= 1#, the curve resembles the curve for #1/x#. With regard to asymptotes, the curve diverges toward #oo# as #xto0# from the positive direction and diverges toward #-oo# as #xto0# from the negative direction.
Outside of that domain, the curve resembles the line #y = x#

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