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# How do you determine whether the function F(x) = x + 1/x is concave up or concave down and its intervals?

Oct 22, 2015

For $f \left(x\right) = x + \frac{1}{x}$, we get $f ' \left(x\right) = 1 - \frac{1}{x} ^ 2$ and $f ' ' \left(x\right) = \frac{2}{x} ^ 3$ which is negative for $x < 0$ and positive for $x > 0$.
So the graph of $f$ is concave down on $\left(- \infty , 0\right)$ and concave up on (0,oo)#
There is no inflection point because the concavity changes at $x = 0$ which is not in the domain of $f$.