Introduction

Local maxima and local minima

Let ‘f’ be a real valued function and let ‘c’ be an interior point in the domain of ‘f’, then
•    ‘c’ is called a point of local maxima if there is an h > 0 such that f(c) > f(x), for all x in (c - h, c + h). The value of f at x = c is called the local maximum value.
•    ‘c’ is called a point of local minima if there is an h > 0 such that f(c) < f(x) for all x in (c - h, c + h). The value of ‘f’ at x = c is called the local minimum value.

Critical Point

A point ‘c’ in the domain of a function ‘f’ at which either f’(c) = 0 or ‘f’ is not differentiable is called a critical point of ‘f’.

First Derivative Test

Let ‘f’ be a function defined on an open interval I. Let ‘f’ be continuous at a critical point ‘c’ in I. Then
•    If f’(x) > 0 at every point sufficiently close to the left of c and f’(x) < 0 at every point sufficiently close to the right of ‘c’ then ‘c’ is a point of local maxima.
•    If f’(x) < 0 at every point sufficiently close to the left of c and f’(x) > 0 at every point sufficiently close to the right of c, then ‘c’ is a point of local minima.
•    If f’(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima.  Such a point is called point of inflection.

Solved Example:
Find all points of local maxima and local minima of the function ‘f’ given by
f(x) = x³ - 3x + 3
f’(x) = 3x² – 3 = 3(x + 1)(x - 1)
f’(x) = 0 implies x = 1 and x = -1

Since the value of f’(x) < 0 to the left of 1 and f’(x) >0 to the right of 1, x = 1 is a point of local minima and f(1) is the local minimum value.
Since the value of f’(x) > 0 to the left of -1 and f’(x) < 0 to the right of -1, x = -1 is a point of local maxima and f(-1) is the local maximum value
Maximum value = 5
Minimum value = 1

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Reference Links:

• http://en.wikipedia.org/wiki/Domain_(ring_theory)
• http://deadline.3x.ro/maxima-minima.html
• http://mathworld.wolfram.com/LocalMinimum.html
• http://en.wikipedia.org/wiki/Inflection_point
• http://en.wikipedia.org/wiki/First_derivative_test