Learning Target: Identify local and absolute extrema using the First Derivative

Question

Learning Target: Identify local and absolute extrema using the First Derivative Test and deter- mine intervals where the function is increasing and decreasing. We saythat a function, f, has a critical point atc if c is in the domain off and f(c)- . f'(c) is . c is an For the graph above identify: 1. Critical Points 2. All local minima 3. All local maxima 4. The absolute minimum (or state if there is not one) 5. The absolute maximum (or state if there is not one) 6. Intervals on which the function is increasing. What is the sign of f'(z) on these intervals? 7. Intervals on which the function is decreasing. What is the sign of f'(x) on these intervals? We say that f is increasing on an interval (a,b), if f[)at We say that f is decreasing on an interval (a,, if e)at each point in (a, b). each point in (a, b).

Explanation / Answer

1) critical points ... , x =0, X = 5.5

Critical points are the points where derivative of the function is zero or tangent to any point is parallel to the x axis.

2).all local minima ...x = -1 , 5.5

3) local Maxima ...x = -3, 0

4). Absolutely minima at x = 5.5

5) absolute Maxima at x = -3

6). Interval in which function is increasing

X = (-5,-3) union (-1,0) union (5.5, infinity)

7). Interval in which function is decreasing

X = ((-3,-1) union (0, 5.5)

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