Deleon Luna
04/02/2024 · High School
Let \( f \) be a function defined by \[ f(x)=\frac{2(x+1)}{x^{2}}, x \neq 0 \] (a) Use the sign pattern for \( f^{\prime}(x) \) to determine the interval(s) over which \( f \) rises and where it falls. (b) Determine the coordinates of the local extreme point(s). (c) Use the sign pattern for \( f^{\prime \prime}(x) \) to determine the interval(s) over which the graph of \( f \) is concave up and (d) Determine the coordinates of the inflection point.
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The function \( f(x) = \frac{2(x+1)}{x^2} \) is increasing on \( (-\infty, -2) \) and \( (-2, 0) \), decreasing on \( (0, \infty) \), has a local maximum at \( (-2, -\frac{1}{2}) \), is concave down on \( (-\infty, -3) \), concave up on \( (-3, 0) \) and \( (0, \infty) \), and has an inflection point at \( (-3, -\frac{4}{9}) \).
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