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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. Your browser does not support the IFRAME tag.
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.

Index

5.1 Tangent And Normal Lines
5.2 Angle Between Two Curves
5.3 Interpretation Of The Sign Of The Derivative
5.4 Locality Increasing Or Decreasing Functions 5.5 Critical Points
5.6 Turning Points
5.7 Extreme Value Theorem
5.8 The Mean-value Theorem
5.9 First Derivative Test For Local Extrema
5.10 Second Derivative Test For Local Extrema
5.11 Stationary Points
5.12 Concavity And Points Of Inflection
5.13 Rate Measure (distance, Velocity And Acceleration)
5.14 Related Rates
5.15 Differentials : Errors And Approximation 