Question

How do you sketch the graph `y=x+1/x` using the first and second derivatives?

Answer 1

See explanation that includes graph that is a rectangular hyperbola. with asymptotes x = 0 and y = 1.

Explanation:

`y'=1-1/x^2=0, when x=+-1.`

`y''=2/x^3 =-2`, at x =`-1` and `2` at x = 1#

Interestingly, no higher derivative is 0 at `x=+-1`..

So, there are no maxima and minima for y. \

As `x to +- oo, y to+-oo`.

As `y to 1, x to+-oo`.

So, x =0 and y = 1 are the asymptotes.

graph{xy-x-1=0 [-10, 10, -5, 5]}