Properties of Functions
Local Minima and Maxima
Def. A function f(x) has a local minimum at a point c if on some open interval, say P, containing c and contained in the domain of f(x), we have:
f(x) ≥ f(c) for all x in P
f(c) is called a local minimum value.
Def. A function f(x) has a local maximum at a point c if on some open interval, say P, containing c and contained in the domain of f(x), we have:
f(x) ≤ f(c) for all x in P
f(c) is called a local maximum value.
In other words, a x=c is a local minimum if (c,f(c)) is locally, around c, the lowest point on the graph. x=c is a local maximum if (c,f(c)) is locally, around c, the highest point on the graph.
The function whose graph you see below has a local minimum at x=1 and a local maximum at x=-1.
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