5.9 First Derivative Test For Local Extrema
If the derivative of a function changes its sign while passing through a critical point along a given curve (i.e. around the critical point), the function possesses a Local (relative) Extrema at that point.
(I) If the derivative changes its sign from positive (increasing function) to negative (decreasing function), the function has a Local maxima at that critical point
(II) If the derivative changes its sign from negative (decreasing function) to positive (increasing function), the function has a Local minima at that critical point.
Technique
1) Find f '(x)
2) Find roots of f '(x) = 0
i.e. critical points of f (x)
3) Let x = c be a critical point of f (x) find f '(c  h) and f '(c +h), h >0
(however small)
(I) If f '(c  h), h >0 and f ' (c + h) <0
Then 'f ' has a Local maximum at x = c
i.e. f (c) = Local maximum
(II) If f ' (c + h) <0 and f '(c + h) >0
Then 'f ' has a Local minimum at x = c
i.e. f (c) = Local minimum
4) Use the same procedure for the other roots.
Note : There is no guarantee that the
derivative will change signs, therefore it is essential to test
each Interval around a critical point.
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