Monotonicity and Extrema of Functions

Overview

This lecture covers the concept of monotonicity and extrema (maximum and minimum) of functions, their relationship to derivatives, how to determine intervals of increase and decrease, and how to analyze critical points using variation tables, sign charts, and function graphs.

Monotonicity of Functions

• Monotonicity refers to intervals where functions are either increasing or decreasing.
• A function is increasing on an interval if ( f'(x) > 0 ) for all ( x ) in that interval.
• A function is decreasing on an interval if ( f'(x) < 0 ) for all ( x ) in that interval.
• If ( f'(x) = 0 ) for all ( x ) in the domain, the function is constant (unchanging).
• Monotonic intervals are determined by the sign of ( f'(x) ).
• The variation table shows how the sign of ( f'(x) ) determines whether ( f(x) ) goes up (increase) or down (decrease).

Critical Points and Extrema

• Critical points occur where ( f'(x) = 0 ) or ( f'(x) ) is undefined.
• An extremum (maximum or minimum) occurs at a point if ( f'(x) ) changes sign across that point and ( f(x) ) is defined there.
• If ( f' ) changes from positive to negative, it’s a local maximum; from negative to positive, it’s a local minimum.
• At a critical point, ( f'(x) ) can be zero or undefined.
• For polynomial functions, extrema occur where ( f'(x) = 0 ).

Variation Table Method

• Steps:
  1. Find the domain of the function.
  2. Compute ( f'(x) ).
  3. Solve ( f'(x) = 0 ) and determine where ( f'(x) ) is undefined.
  4. Create a variation table with these points, checking the sign of ( f'(x) ) in each interval.
  5. Conclude increasing/decreasing intervals and extrema locations using the sign changes.

Graphical Interpretation

• For ( f(x) ): convex points are maxima, concave points are minima.
• For ( f'(x) ):
  • Where graph crosses the x-axis, ( f'(x) = 0 ) (potential extrema for ( f(x) )).
  • ( f'(x) > 0 ) above x-axis (function increasing), ( f'(x) < 0 ) below (function decreasing).
• Double roots (even powers) in ( f'(x) ) do not produce extrema, as the sign does not change.

Key Terms & Definitions

• Monotonicity — The property of being exclusively increasing or decreasing on an interval.
• Critical Point — A point where ( f'(x) = 0 ) or ( f'(x) ) is undefined.
• Extremum (pl. Extrema) — A maximum or minimum value of a function.
• Variation Table — A table showing the sign of ( f'(x) ) and behavior of ( f(x) ) over intervals.
• Constant Function — A function where ( f'(x) = 0 ) everywhere.

Action Items / Next Steps

• Practice identifying intervals of increase/decrease and extrema using derivatives and variation tables.
• Solve assigned textbook problems on monotonicity and extrema as demonstrated.
• Review how to interpret function and derivative graphs for monotonicity and extrema.