Question

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

Answer 1

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`