# Other Concepts

*Maxima* and *minima* are high and low spots on a function, respectively. Collectively, they are referred to as *extrema*. An extreme can be either *local* (the highest/lowest spot on an interval) or *absolute* (the highest/lowest point in the whole function).

Extrema are easy to find with calculus; they always occur where the first directive is equal to 0.

*Concavity* refers to which way a function opens on some interval. A function can be either *concave up* or *concave down*. A point where concavity changes is known as an *inflection point*.

A function, ** f(x)**, is concave up on an interval if and only if its first derivative,

**is increasing on that interval (i.e.**

*f’*(x)**).**

*f”*(x) > 0A function, ** f(x)**, is concave down on an interval if and only if its first derivative,

**is decreasing on that interval (i.e.**

*f’*(x)**).**

*f”*(x) < 0Inflection points occur on a function wherever the second derivative is equal to 0 (** f”(x) = 0**).

## Sources

Shenk, A. (1988). *Calculus and Analytic Geometry* [4th Ed.]. Glenview, IL: Scott Foresman and Company.

Swokowski, E., Olinick, M., Pence, D. & Cole, J. (1994). *Calculus of a Single Variable* [6th Ed.]. Boston, MA: PWS.

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