1. **Problem Statement:** We need to find the sign diagram for the derivative of the function $$y = x + \frac{1}{x}$$. 2. **Find the derivative:** Use the power rule and the derivative of $$\frac{1}{x}$$ which is $$x^{-1}$$. $$y' = \frac{d}{dx}\left(x + x^{-1}\right) = 1 - x^{-2} = 1 - \frac{1}{x^2}$$ 3. **Simplify the derivative:** $$y' = \frac{x^2 - 1}{x^2}$$ 4. **Determine critical points:** Set numerator equal to zero: $$x^2 - 1 = 0 \implies x^2 = 1 \implies x = \pm 1$$ 5. **Analyze sign of $$y'$$:** - The denominator $$x^2$$ is always positive except at $$x=0$$ where the function is undefined. - The sign of $$y'$$ depends on the numerator $$x^2 - 1$$. 6. **Sign intervals:** - For $$|x| > 1$$, $$x^2 - 1 > 0$$ so $$y' > 0$$. - For $$|x| < 1$$, $$x^2 - 1 < 0$$ so $$y' < 0$$. - At $$x=0$$, $$y'$$ is undefined. 7. **Summary:** - $$y' > 0$$ on $$(-\infty, -1) \cup (1, \infty)$$ - $$y' < 0$$ on $$(-1, 0) \cup (0, 1)$$ - Critical points at $$x = -1$$ and $$x = 1$$ - Vertical asymptote at $$x=0$$ This sign diagram shows where the function is increasing or decreasing based on the derivative's sign.