Graph of a Function Def. A function f (x) has a local maximum (relative max) at x = p if f (x) < f (p) for all points near p. Def. A function f (x) has.

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Presentation transcript:

Graph of a Function Def. A function f (x) has a local maximum (relative max) at x = p if f (x) < f (p) for all points near p. Def. A function f (x) has a local minimum (relative min) at x = p if f (x) > f (p) for all points near p.

Def. We say that p is a critical point of f (x) if f (p) = 0 or is undefined. Thm. All local max./min. points of a function are critical points.  The converse is not true.

First Derivative Test If f (x) is positive before p and negative after p, then p is a local maximum. If f (x) is negative before p and positive after p, then p is a local minimum.

Ex. Find and classify all critical points of f (x) = x3 – 5x2 + 3x – 1.

Def. We say that p is an inflection point of f (x) if the concavity of f changes at p. Thm. If p is an inflection point of f (x), then f (p) = 0.  The converse is not true.

Ex. Find all inflection points of

Ex. Identify the x-coordinate of all points where f (x) has a local max., local min., and inflection point.

Ex. The table contains selected values of differentiable function f (x) and its derivative. Explain why it’s not possible to determine the x-coordinates of local extrema.

Ex. If f (x) = ax2 + bx, find values of a and b that would result in a local max at (1,5).