# Basic Definitions of Asymptotes, Concavity, Critical Points Etc

Asymptotes

ConcavityDefinition of a horizontal asymptote:The line y = y_{0}is a "horizontal asymptote" of f(x) if and only if f(x) approaches y_{0}as x approaches + or - .Definition of a vertical asymptote:The line x = x_{0}is a "vertical asymptote" of f(x) if and only if f(x) approaches + or - as x approaches x_{0}from the left or from the right.Definition of a slant asymptote:the line y = ax + b is a "slant asymptote" of f(x) if and only if lim_{(x–>+/-)}f(x) = ax + b.

Critical PointsDefinition of a concave up curve:f(x) is "concave up" at x_{0}if and only iff ‘(x) is increasing at x_{0}Definition of a concave down curve:f(x) is "concave down" at x_{0}if and only iff ‘(x) is decreasing at x_{0}The second derivative test:If f”(x) exists at x_{0}and is positive, thenf “(x) is concave up at x_{0}. If f”(x exists and is negative, then f(x) is concave down at x_{0})_{0}. Iff “(x) does not exist or is zero, then the test fails.

Extrema (Maxima and Minima)Definition of a critical point:a critical point on f(x) occurs at x_{0}if and only if either f ‘(x_{0}) is zero or the derivative doesn’t exist.

Increasing/Decreasing FunctionsLocal (Relative) ExtremaDefinition of a local maxima:A function f(x) has a local maximum at x_{0}if and only if there exists some interval I containing x_{0}such that f(x_{0}) >= f(x) for all x in I.Definition of a local minima:A function f(x) has a local minimum at x_{0}if and only if there exists some interval I containing x_{0}such that f(x_{0}) <= f(x) for all x in I.Occurrence of local extrema:All local extrema occur at critical points, but not all critical points occur at local extrema.The first derivative test for local extrema:If f(x) is increasing (f ‘(x) > 0) for all x in some interval (a, x_{0}] and f(x) is decreasing (f ‘(x) < 0) for all x in some interval [x_{0}, b), then f(x) has a local maximum at x_{0}. If f(x) is decreasing (f ‘(x) < 0) for all x in some interval (a, x_{0}] and f(x) is increasing (f ‘(x) > 0) for all x in some interval [x_{0}, b), then f(x) has a local minimum at x_{0}.The second derivative test for local extrema:Iff ‘(x = 0 and_{0})f “(x > 0, then f(x) has a local minimum at x_{0})_{0}. Iff ‘(x = 0 and_{0})f “(x < 0, then f(x) has a local maximum at x_{0})_{0}.Absolute ExtremaDefinition of absolute maxima:y_{0}is the "absolute maximum" of f(x) on I if and only if y_{0}>= f(x) for all x on I.Definition of absolute minima:y_{0}is the "absolute minimum" of f(x) on I if and only if y_{0}<= f(x) for all x on I.The extreme value theorem:If f(x) is continuous in a closed interval I, then f(x) has at least one absolute maximum and one absolute minimum in I.Occurrence of absolute maxima:If f(x) is continuous in a closed interval I, then the absolute maximum of f(x) in I is the maximum value of f(x) on all local maxima and endpoints on I.Occurrence of absolute minima:If f(x) is continuous in a closed interval I, then the absolute minimum of f(x) in I is the minimum value of f(x) on all local minima and endpoints on I.Alternate method of finding extrema:If f(x) is continuous in a closed interval I, then the absolute extrema of f(x) in I occur at the critical points and/or at the endpoints of I.

(This is a less specific form of the above.)

Inflection PointsDefinition of an increasing function:A function f(x) is "increasing" at a point x_{0}if and only if there exists some interval I containing x_{0}such that f(x_{0}) > f(x) for all x in I to the left of x_{0}and f(x_{0}) < f(x) for all x in I to the right of x_{0}.Definition of a decreasing function:A function f(x) is "decreasing" at a point x_{0}if and only if there exists some interval I containing x_{0}such that f(x_{0}) < f(x) for all x in I to the left of x_{0}and f(x_{0}) > f(x) for all x in I to the right of x_{0}.The first derivative test:Iff ‘(x exists and is positive, then_{0})f ‘(x) is increasing at x_{0}. Iff ‘(x) exists and is negative, then f(x) is decreasing at x_{0}. Iff ‘(x does not exist or is zero, then the test tells fails._{0})

Definition of an inflection point:An inflection point occurs on f(x) at x_{0}if and only if f(x) has a tangent line at x_{0}and there exists and interval I containing x_{0}such that f(x) is concave up on one side of x_{0}and concave down on the other side