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# 4.4.8-eg - Example Consider the curve y = f(x where f(x =...

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Example Consider the curve y = f ( x ), where f 0 ( x ) = 4 x 2 / 5 . Find the open intervals where f is increasing and, decreasing, and where f has relative (local) extrema. Also ﬁnd the intervals where where f is concave up and concave down, and where f has point(s) of inﬂection. Then ﬁnd the y -intercept and x -intercept(s) of the curve, and any vertical asymptotes. Determine the y - coordinates associated with locations of local extrema and point(s) of inﬂection, if any, and use this information to sketch the graph of the curve. Solution: The domain of the function f is ( -∞ , ). Since f 0 ( x ) = 4 · 2 5 x - 3 / 5 = 8 5 x 3 / 5 it follows that f 0 ( x ) is undeﬁned at x = 0 and that f 0 ( x ) is never zero. Therefore, there is only one critical point, at x = 0 . A sign chart for f 0 follows: interval f 0 ( x ) = 8 5 x 3 / 5 behavior of f ( -∞ , 0) - decreasing (0 , ) + increasing Thus: f is increasing on (0 , ), f is decreasing on ( -∞ , 0). Applying the First Derivative Test to the critical point

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4.4.8-eg - Example Consider the curve y = f(x where f(x =...

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