Suppose you are given a formula for a function f. (a) How do you determine where f is increasing or decreasing? If f'(x) ?0 on an interval, then f is increasing on that interval. If f'(x) ? o on an interval, then f is decreasing on that interval. (b) How do you determine where the graph of f is concave upward or concave downward? If f"(x) 2o for all x in I, then the graph of f is concave upward on I. If f"(x) ? |O for all x in I, then the graph of f is concave downward on I. (c) How do you locate inflection points? At any value of x where the concavity changes, we have an inflection point at (x, f(x)). At any value of x where the function changes from increasing to decreasing, we have an inflection point at (x, f(x)). At any value of x where f'(x) = 0, we have an inflection point at (x, f(x)). At any value of x where the function changes from decreasing to increasing, we have an inflection point at (x, f(x)). At any value of x where the concavity does not change, we have an inflection point at (x, f(x)).

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