Finding Intervals of Increase/Decrease - University of Wisconsin–La ...

[Pages:1]Computing Intervals of Increase/Decrease and Intervals of Concavity (E. Kim)

Finding Intervals of Increase/Decrease

1. Draw a number line for x-values (labeled with f ) 2. Draw a "breakpoint" for each x-value c where f (c) does not exist. 3. Draw a "breakpoint" for each x-value c where f (c) = 0 or f (c) does not exist. In other words, draw

in a breakpoint for each critical number. (This includes places where f (c) does not exist, labeled in the previous step.) 4. Pick a number x in each interval separated by breakpoints and compute f (x).

? If f (x) > 0, then f is increasing in this interval. ? If f (x) < 0, then f is decreasing in this interval. ? If f (x) = 0, then f is constant in this interval. Remark: Because f (c) does not exist if f (c) does not exist, listing step 2 is actually redundant.

Finding Intervals of Concavity

1. Draw a number line for x-values (labeled with f ) 2. Draw a "breakpoint" for each x-value c where f (c) does not exist. 3. Draw a "breakpoint" for each x-value c where f (c) does not exist. 4. Draw a "breakpoint" for each x-value c where f (c) = 0 or f (c) does not exist. 5. Pick a number x in each interval separated by breakpoints and compute f (x).

? If f (x) > 0, then f is concave up (CU) in this interval. ? If f (x) < 0, then f is concave down (CD) in this interval. Remark: Similarly to the previous remark, because f (c) does not exist if f (c) does not exist, listing step 3 is actually redundant. In turn, listing step 2 is also redundant.

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