Absolute (global) extrema. For f defined on a domain D, f(x0) is the absolute maximum of f on D if f(x0)≥f(x) for every x∈D; the absolute minimum is defined symmetrically with ≤.
Extreme Value Theorem. If f is continuous on a closed interval [a,b], then f attains both an absolute maximum and an absolute minimum somewhere on [a,b] — and the extremum can only occur either at an interior critical number or at one of the two endpoints.
Procedure for absolute extrema on [a,b] (Exercise 7.6 Q1's method):
- Find every critical number of f in the open interval (a,b).
- Evaluate f at each critical number and at both endpoints a,b.
- The largest of these values is the absolute maximum; the smallest is the absolute minimum.
Relative (local) extrema. f has a relative (local) maximum at x0 if f(x0) is the largest value of f on some open interval around x0 (relative minimum: smallest, on some open interval). A function may have several local extrema, and a local extremum need not be the absolute one.
Fermat's Theorem. If f has a relative extremum at x=c, then c must be a critical number of f (so the search for local extrema always starts by solving f′(x)=0 together with any points where f′ fails to exist) — though not every critical number is automatically an extremum (e.g. y=x3 at x=0).
First Derivative Test. At a critical point c where f is continuous, examine the sign of f′(x) moving left to right across c:
- negative → positive: local minimum at c;
- positive → negative: local maximum at c;
- no sign change (same sign on both sides): c is neither a local max nor a local min.
Second Derivative Test (an alternative, often quicker, at a stationary point). If f′(c)=0 and f′′(c) exists:
- f′′(c)<0 ⇒ local maximum at c;
- f′′(c)>0 ⇒ local minimum at c; …