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 - Examples of Local and Absolute Maxima and Minima

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.

Concavity and Inflection Points

A function, f(x), is concave up on an interval if and only if its first derivative, f’(x) is increasing on that interval (i.e. f”(x) > 0).

A function, f(x), is concave down on an interval if and only if its first derivative, f’(x) is decreasing on that interval (i.e. f”(x) < 0).

Inflection 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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