Overview
This lecture covers how to find local (relative) maximum and minimum values of a function using derivatives, including critical numbers, sign charts, and examples.
Identifying Local Extrema with Derivatives
- Local maxima and minima occur where the derivative equals zero (horizontal tangent line).
- These points are called critical numbers.
- The first step is to find the derivative of the given function and set it to zero.
- Solve for x to find the critical numbers.
- Use a sign chart to determine whether each critical number corresponds to a local max or min.
Example 1: ( f(x) = x^2 - 4x )
- The derivative is ( f'(x) = 2x - 4 ).
- Set ( 2x - 4 = 0 ); solve to get ( x = 2 ).
- Test intervals on the sign chart: derivative is negative before 2, positive after.
- Conclusion: Local minimum at ( x = 2 ).
- Value: ( f(2) = 4 - 8 = -4 ), so the minimum is at (2, -4).
Example 2: ( f(x) = 2x^3 + 3x^2 - 12x )
- The derivative is ( f'(x) = 6x^2 + 6x - 12 ).
- Factor: ( 6(x+2)(x-1) ).
- Critical numbers are ( x = -2 ) and ( x = 1 ).
- Use a sign chart for intervals around -2 and 1.
- At ( x = -2 ), slope goes from positive to negative: local maximum.
- At ( x = 1 ), slope goes from negative to positive: local minimum.
- Values: ( f(-2) = 20 ) (max), ( f(1) = -7 ) (min).
Example 3: ( f(x) = 3x^4 - 16x^3 + 24x^2 )
- The derivative is ( f'(x) = 12x^3 - 48x^2 + 48x ).
- Factor: ( 12x(x-2)^2 ).
- Critical numbers are ( x = 0 ) and ( x = 2 ).
- Use sign chart; at ( x = 0 ), the sign changes (min), at ( x = 2 ) it does not (neither min nor max).
- Conclusion: Only a minimum at ( x = 0 ).
Key Terms & Definitions
- Local (Relative) Maximum/Minimum — Highest/lowest point in a small region around the point.
- Critical Number — Value of x where the derivative is zero or undefined.
- First Derivative Test — Method using sign changes of the derivative to identify local extrema.
- Sign Chart — Diagram showing the sign of the derivative on intervals between critical numbers.
- Multiplicity — The exponent of a factor; odd multiplicity switches sign, even keeps sign.
Action Items / Next Steps
- Practice finding local extrema for different polynomial functions.
- Review sign charts and first derivative test for various types of critical points.