Calculus

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) _________ 0 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) _____ for all x in I, then the graph of f is concave upward on I.

If f"(x) _______ 0 for all x in I, then the graph of f is concave downward on I.

(c) How do you locate inflection points?

a) At any value of x where the function changes from increasing to decreasing, we have an inflection point at (x, f(x)). b) At any value of x where the concavity changes, we have an inflection point at (x, f(x)). c) At any value of x where the function changes from decreasing to increasing, we have an inflection point at (x, f(x)). d) At any value of x where the concavity does not change, we have an inflection point at (x, f(x)). e) At any value of x where f'(x) = 0, we have an inflection point at (x, f(x)).

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