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Critical point
From Mathematics Is A Science
Revision as of 17:45, 7 August 2010 by imported>WikiSysop
How do we find extreme points of functions of one variable?
Let y = f(x). To find the local maximum or minimum, proceed as follows:
- compute f′(x),
- solve f′(x) = 0 for x, (also find x's for which f′(x) does not exist)
- these are the critical points of f.
- some of these are maxima, some are minima, others are neither (in this case, they will be called "saddles").
- 2nd derivative test: classify the critical points based on the sign of f(a). Note that it is possible that a is a minimum and f′′(a) = 0. Example: y = x4 at a = 0.
- 1st derivative test: classify the critical points based on change of the sign of f'(a) f′(x) > 0 for x < a and f′(x) < 0 for x > a, f′(x) < 0 for x < a and f′(x) > 0 for x > a, no change.