Suppose you are given a formula for a function f (a) How do you determine where

Question

Suppose you are given a formula for a function f (a) How do you determine where f is increasing or decreasing? If x) If rx) on an interval, then f is increasing on that interval on an interval, then f Iis decreasing on that interval. (b) How do you determine where the graph of fis concave upward or concave downward? If "x)o for all x in I, then the graph of f is concave upward on . 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? O At any value of x where the concavity does not change, we have an inflection point at (x, f(x)). O At any value of x where the concavity changes, we have an inflection point at (x, (x)). O At any valu? of x where the function changes from increasing to decreasing, we have an inflection point at (x, x)). At any value of x where f"(x) = 0, we have an infection point at (x, rx)). O At any value of x where the function changes from decreasing to increasing, we have an inflection point at (x, (x)). Need Help? ReadtTak to s Tutor 4.-6 points SC The graph of the first derivative f' of a function f is shown. (Assume the function is defmed only for o sx s 9.)

Explanation / Answer

Increase or Decrease:

Let f be a function that is continuous on a closed interval [a, b], and differentiable on the open interval (a, b). (a) If f '(x) > 0 for every value of x in (a, b), then f is increasing in [a, b].

(b) If f '(x) < 0 for every value of x in (a, b), then f is decreasing in [a, b].

(c) If f ' (x) = 0 for every value of x in (a, b), then f is constant on [a, b].

concave up or concave down :

Let f be twice differentiable on an open interval I.

(a) If f'' (x) > 0 on I, then f is concave up on I.

(b) If f'' (x) < 0 on I, then f is concave down on I.

point of inflection:

If f is continuous on an open interval containing the point x0 , and if f changes the direction of its concavity at that point, then we say that f has an inflection point at x0 , and we call the point (x0 , f(x0 )) on the graph of f an inflection point of f.

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