## 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*) at a point where *x* = *c* if for all *x* in the immediate neighborhood of *c*. If a curve has a local at *x* = *c*, then the curve changes from 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* of a function on [*a*, *b*] occurs at *x* = *c* if for all *x* on [*a*, *b*].

A curve is said to be *concave* at a point *P*(*x*_{1}, *y*_{1}) if the curve lies its tangent. If at *P*, the curve is concave 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*.

**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*.