Calculus AB and Calculus BC

CHAPTER 4 Applications of Differential Calculus

D. MAXIMUM, MINIMUM, AND INFLECTION POINTS: DEFINITIONS

The curve of y = f (x) has a local (or relative) Image at a point where x = c if Image for all x in the immediate neighborhood of c. If a curve has a local Image at x = c, then the curve changes from Image as x increases through c. If a function is differentiable on the closed interval [a, b] and has a local maximum or minimum at x = c (a < c < b), then f (c) = 0. The converse of this statement is not true.

If f (c) is either a local maximum or a local minimum, then f (c) is called a local extreme value or local extremum. (The plural of extremum is extrema.)

The global or absolute Image of a function on [a, b] occurs at x = c if Image for all x on [a, b].

A curve is said to be concave Image at a point P(x1, y1) if the curve lies Image its tangent. If Image at P, the curve is concave Image In Figure N4–1, the curves sketched in (a) and (b) are concave downward at P while in (c) and (d) they are concave upward at P.

Image

FIGURE N4–1

Point of inflection

A point of inflection is a point where the curve changes its concavity from upward to downward or from downward to upward. See §I, for a table relating a function and its derivatives. It tells how to graph the derivatives of f, given the graph of f.