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

Nov 26, 2016

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

#### Explanation:

$y ' = 1 - \frac{1}{x} ^ 2 = 0 , w h e n x = \pm 1.$

$y ' ' = \frac{2}{x} ^ 3 = - 2$, at x =$- 1$ and $2$ at x = 1#

Interestingly, no higher derivative is 0 at $x = \pm 1$..

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

As $x \to \pm \infty , y \to \pm \infty$.

As $y \to 1 , x \to \pm \infty$.

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

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