Section 2 8 What do f and f say about f Definition A function f x is increasing if whenever a b f a f b A function is decreasing if whenever a b f b f a If a function is differentiable on a b then the following are true If f 0 on a b then f is increasing of a b If f 0 on a b then f is decreasing on a b If f 0 on a b then f is concave up on a b If f 0 on a b then f is concave down on a b How do we find the signs of f and f Example Given that f x x 2 where is f increasing decreasing x 2 is positive only for x 2 so f is increasing on 2 and decreasing on 2 Example Given that f x x 2 x 3 where is f increasing decreasing We have to find the intervals where f is positive One way is to graph f and see where f is above the x axis x 2 x 3 is a parabola opening upward with zeros at x 2 and x 3 so it is positive on 2 and 3 and this is where f is increasing f is negative on 2 3 and this is where f is decreasing Another way is to make a sign chart for f The sign of f can only change if x 2 changes sign or if x 3 changes sign x a can only change sign at a Each of these does change sign once If we test the sign at x 0 then change at x 2 and change it again at x 3 we get the following sign chart f 0 6 is positive so we start with signf 2 3 In the next example f has a factor that is squared so that factor will not change the sign of f Example Given f x x 2 2 x 3 make a sign chart for f Again x 0 is the most convenient point to test f 0 12 so we start out negative The sign does not change at 2 because x 2 is squared but it does change at 3 sign f 2 3 So f is decreasing on 3 and increasing on 3 x 1 x 2 2 Example f x x 5 3 THe only x values where f can change sign are 1 2 and 5 The factor x 1 does not change sign but x 2 3 and x 5 both do change sign f 0 8 5 is positive so we put a between 1 and 2 where 0 is Then change the signs as we cross over 2 and 5 but not as we cross over 1 sign f 1 2 5
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