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5.10 Second Derivative Test For Local Extrema

Maximum points

Consider the point P (Fig 1). The gradient at P is zero for the curve y = f(x). The gradient is positive for all points to the immediate left of P and negative for all points to the immediate right of P.

Thus near P is changing its sign from positive, through zero to negative values, therefore at P, i.e. is negative.

Therefore, for maximum point,and

Minimum points

Consider the point P. The gradient there is zero. For all points to the immediate right of P, the gradient is positive while for all points to the immediate left of P, the gradient is negative. Thus near P, changes from negative, through zero to positive values.

Therefore, at is positive.

Therefore,for a minimum point

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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

Chapter 6





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