Quiz 7, Math110 Name: Student ID: Write Down detailed steps. Partial credits will be given for that. 1. Find the interval(s) where the function f(x) = 2x3 + 12x2 ¡7x¡11 is increasing and where it is decreasing. Solution: f0(x) = 6x2 +x¡7 ) f0(x) = (6x+7)(x¡1) ) x = ¡76 or 1 (¡1;¡76) ) f0(x) are always positive. (¡76;1) ) f0(x) are always negative. (¡1;+1) ) f0(x) are always positive. So, it?s easy to get the conclusion that increasing: (¡1;¡76)[(¡1;+1), decreasing(¡76;1). 2. Find the relative maxima and relative minima of the function. f(x) = 2x¡13x+1 Solution: f0(x) = 5(3x+1)2 It?s easy to check that f0(x) are always positive, which means there is no change of sign for f0(x), namely, the relative extreme does not exist. 3. Determine where the function f(x) is concave upwards and concave downwards. f(x) = x3 +2x2 ¡3x+1 Solution: f0(x) = 3x2 +4x¡3 f00(x) = 6x+4 So, when x > ¡23, f0(x) is positive, it?s concave upwards. For x < ¡23, f0(x) is negative, it?s concave downwards.
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