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

1 Answer
Douglas K.
Nov 5, 2016

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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