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

Please see the explanation.

Explanation:

Here is a graph of the curves:

graph{x+1/x}

The first derivative is:

$\frac{\mathrm{dy}}{\mathrm{dx}} = 1 - \frac{1}{x} ^ 2$

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

$1 - \frac{1}{x} ^ 2 = 0$

$1 = \frac{1}{x} ^ 2$

${x}^{2} = 1$

$x = \pm 1$

$y \left(- 1\right) = - 1 + \frac{1}{-} 1 = - 2$

$y \left(1\right) = 1 + \frac{1}{1} = 2$

Perform the second derivative test:

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = \frac{1}{x} ^ 3$

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

This makes the point $\left(1 , 2\right)$ a local minimum and the point $\left(- 1 , 2\right)$ a local maximum. Within domain $- 1 \le x \le 1$, the curve resembles the curve for $\frac{1}{x}$. With regard to asymptotes, the curve diverges toward $\infty$ as $x \to 0$ from the positive direction and diverges toward $- \infty$ as $x \to 0$ from the negative direction.
Outside of that domain, the curve resembles the line $y = x$

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