Finding Absolute Maximum And Minimums Homework For Kids
Definition of a concave down curve: f(x) is "concave down" at x_{0} if and only if f '(x) is decreasing at x_{0}
The second derivative test: If f ''(x) exists at x_{0} and is positive, then f ''(x) is concave up at x_{0}. If f ''(x_{0}) exists and is negative, then f(x) is concave down at x_{0}. If f ''(x) does not exist or is zero, then the test fails.
Definition 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: If f '(x_{0}) = 0 and f ''(x_{0}) > 0, then f(x) has a local minimum at x_{0}. If f '(x_{0}) = 0 and f ''(x_{0}) < 0, then f(x) has a local maximum at x_{0}.
Absolute Extrema
Definition 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.)
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: If f '(x_{0}) exists and is positive, then f '(x) is increasing at x_{0}. If f '(x) exists and is negative, then f(x) is decreasing at x_{0}. If f '(x_{0}) does not exist or is zero, then the test tells fails.
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