Mathematics

Mathematics

Problem

Consider the function ff whose derivative is f′(x)=(x−1)2(x+4)f'(x)=(x-1)^2(x+4). Which of the following statements is correct? a) ff is decreasing on (−∞,1)(-\infty,1), increasing on (1,∞)(1,\infty), attains a relative minimum at x=1x=1, and is concave down on (−73,∞)\left(-\dfrac{7}{3},\infty\right). b) ff is decreasing on (−∞,−4)(-\infty,-4), increasing on (−4,∞)(-4,\infty), attains a relative minimum at x=−4x=-4, and is concave down only on (−73,1)\left(-\dfrac{7}{3},1\right). c) ff is decreasing on (−∞,−4)(-\infty,-4), increasing on (−4,∞)(-4,\infty), attains a relative minimum at x=−4x=-4, and is concave down on (−∞,−73)∪(1,∞)(-\infty,-\dfrac{7}{3})\cup(1,\infty). d) ff is increasing on (−∞,−4)(-\infty,-4), decreasing on (−4,∞)(-4,\infty), attains a relative maximum at x=−4x=-4, and is concave down only on (−∞,−73)(-\infty,-\dfrac{7}{3}).

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