Second Derivative Test

Introduction

• Purpose: Understand and intuitively grasp the second derivative test.
• Visual Aid: Use of x and y axes to depict scenarios.

Key Concepts

Local Maximum

• Visualization: Point ( c ) is a local maximum with function ( f(c) ).
• Characteristics:
  • Tangent slope at ( x = c ) is zero: ( f'(c) = 0 ).
  • Concave downward around ( x = c ):
    • Slope decreases from positive, to zero, to negative.
    • Second derivative is less than zero: ( f''(c) < 0 ).

Local Minimum

• Visualization: Point ( c ) is a local minimum.
• Characteristics:
  • Tangent slope at ( x = c ) is zero: ( f'(c) = 0 ).
  • Concave upwards around ( x = c ):
    • Slope increases from zero to positive.
    • Second derivative is greater than zero: ( f''(c) > 0 ).

Second Derivative Test Explained

• Function Requirements: Twice differentiable function ( f ).
• Test Steps:
  1. Determine ( f'(c) = 0 ) and ( f'' ) exists in a neighborhood around ( c ).
  2. Assess ( f''(c) ):
     • ( f''(c) < 0 ): Relative maximum at ( x = c ).
     • ( f''(c) > 0 ): Relative minimum at ( x = c ).
     • ( f''(c) = 0 ): Inconclusive.

Example Application

• Function: ( h ) is twice differentiable.
• Given:
  • ( h(8) = 5 )
  • ( h'(8) = 0 )
  • ( h''(8) = 4 )
• Determine: Nature of point ( (8, 5) ).
  • Analysis:
    • ( h'(8) = 0 ): Potential critical point.
    • ( h''(8) > 0 ): Relative minimum at ( (8, 5) ).

Conclusion

• Using the second derivative test allows determination of relative maximum or minimum based on the sign of the second derivative when the first derivative is zero.
• Important Note: If second derivative is zero, result is inconclusive.