group hw 7 - (4/1 36 It is given that x=3 is the critical...

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(4/1) 36 . It is given that x=3 is the critical value meaning at x=3 the slope is 0. Looking at the example derivative graph that plots the given derivatives we can see that, where x is less than 3, the slope of f(x) is greater than 0 since the values are positive. Also, when x is greater than 3, the slope of f(x) is less than 0 since the values are negative. Thus, the function would be increasing before the point f (3) and decreasing afterwards. Hence, at x=3 we find the local maximum. See possible graph of original and derivative equations below. b) We can tell that x=3 is the local minimum because as x approaches infinity and negative infinity, the limits are both infinity. So, with one critical point, we know that this graph must be a parabola and x=3 must be a minimum value (since the values on either side approach infinity). See possible original function (a parabola) below. c) The slopes before and after x=3 are positive. In order to change slope direction there has to be a critical point.
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