Continuous Functions, Extreme Values, and Tangent Lines at Local Extrema

Extreme Value Guarantees on Closed Bounded Intervals

• Conditions for Guaranteed Absolute Extrema:
  • When a function is continuous on a closed, bounded interval (an interval with two endpoints where both are included within brackets, represented as $[a, b]$), the existence of both an absolute maximum and an absolute minimum is guaranteed.
• Implications When Conditions Are Not Met:
  • If the interval is not closed and bounded, or if the function is not continuous, the guarantee of absolute extrema ceases to apply.
  • The absence of a guarantee does not mean that an absolute maximum or minimum cannot exist; rather, it means their presence is not assured.
  • Without these conditions, arbitrary outcomes can arise (such as behaviors seen with Swedenborg or parabolas).

Geometric and Intuitive Nature of Local Extrema

• Intuition of a Local Maximum:
  • A local maximum functions as a local high spot, directly comparable to a mountain top.
  • Leading up to this point, the curve ascends as the function increases.
  • Once the peak or mountain top is crossed, continuing past it results in a descent as the function values decrease.
  • The point where the path transitions from ascending to descending is where the graph turns around.

Behavior of Tangent Lines and Derivatives at Smooth Extrema

• Differentiability on Smooth Curves:
  • If the graph is smooth at the turning point, the derivative exists at that location.
  • The existence of the derivative directly corresponds to the existence of a well-defined tangent line.
• Orientation of the Tangent Line:
  • All tangent lines are straight lines.
  • At the exact turning point (peak) of a smooth curve, the tangent line is oriented horizontally.
  • A horizontal tangent line corresponds to a derivative value of $0$.

Non-Smooth Extrema and Sharp Points

• Behavior at Non-Smooth Points:
  • At a sharp point (such as a corner or cusp), there is no horizontal tangent line because no tangent line exists at all.
  • The derivative fails to exist at these sharp locations.
• Symmetry with Local Minima:
  • The identical principles govern local minima: a smooth valley produces a horizontal tangent line, whereas a sharp bottom point lacks a tangent line entirely.
• Irregular and Pathological Curves:
  • Curves can present conditions significantly more irregular and complicated than simple sharp points, leading to a wide variety of non-differentiable behaviors at extrema.

Questions & Discussion

• Tangent Line Orientation at a Smooth Extremum:
  • Question: At this turning point on a smooth graph, how does the tangent line go?
  • Response: It is horizontal; there is a horizontal tangent line.
• Tangent Line Existence at a Sharp Point:
  • Question: Do we have a horizontal tangent line at this sharp point?
  • Response: No, because there is no tangent line at a sharp point.
• Audience Clarifications:
  • Question: Any questions about this one?
  • Response: Next example.