Extrema on Closed and Bounded Regions

Concept Explanation

The Multi-Tiered Candidate Search Strategy

Because absolute extrema on a compact domain $D$ can reside either strictly inside the interior or along the boundary edges, engineers must execute a structured 3-Tier Search Strategy:

        Boundary Edge 2: y = 4 (0 <= x <= 4)
     (0, 4) +-----------------------+ (4, 4)
            |                       |
            |      * Interior       |
Boundary    |        Critical Point | Boundary Edge 3:
Edge 1:     |        (x_0, y_0)     | x = 4 (0 <= y <= 4)
x = 0       |                       |
            |                       |
     (0, 0) +-----------------------+ (4, 0)
        Boundary Edge 4: y = 0 (0 <= x <= 4)

1. Tier 1: Interior Search

   • Compute first partial derivatives $f_x$ and $f_y$ and set them to zero:
     $$\nabla f(x, y) = \mathbf{0} \implies f_x = 0, \quad f_y = 0$$
   • Domain Filtering Rule: Retain only those critical points that lie strictly inside the open interior of $D$. Any critical point lying outside $D$ must be discarded.
   • Note: You do not need to compute the second derivative test discriminant $D$ here; you only need the function values $f(x, y)$ at the candidate points.

2. Tier 2: Boundary Edge Search

   • Decompose the boundary $\partial D$ into smooth parameterized curves or line segments (e.g., straight sides of a polygon, circular arcs).
   • For each boundary segment, express the boundary mathematically:
     • By direct substitution: Replace one variable with an expression in terms of the other (e.g., $y = mx + b$ or $y^2 = R^2 - x^2$), reducing $f(x, y)$ to a single-variable function $g(x)$.
     • By parameterization: Express the curve as $\vec{r}(t) = \langle x(t), y(t) \rangle$ for $t \in [t_1, t_2]$, giving $g(t) = f(x(t), y(t))$.
   • Differentiate $g'(x) = 0$ (or $g'(t) = 0$) to locate 1D critical points along that boundary segment, verifying they lie within the segment interval.

3. Tier 3: Corner / Vertex Check

   • Identify all intersection points (vertices or corners) where adjacent boundary segments meet (e.g., $(0,0), (4,0), (0,3)$ on a triangle).
   • Boundary derivatives may not be defined across sharp corners, so all vertices are automatically candidate points.

4. Global Evaluation & Comparison

   • Construct a master evaluation table listing every candidate point from Tiers 1, 2, and 3.
   • Evaluate the original function $f(x, y)$ at every candidate.
   • Absolute Maximum: The largest value in the table.
   • Absolute Minimum: The smallest value in the table.

Engineering Applications

• Civil & Geotechnical Engineering: Foundation footing slabs transmit vertical column loads to underlying subgrade soil. Geotechnical engineers determine the absolute maximum soil bearing pressure $P(x, y)$ across triangular or rectangular footings to ensure values do not exceed the soil's ultimate bearing capacity $q_{\text{ult}}$, preventing catastrophic footing shear failure.
• Mechanical & Materials Engineering: Silicon microprocessor dies operate under high thermal power dissipation. Mechanical engineers calculate temperature fields $T(x, y)$ on closed rectangular dies ($0 \le x \le L, 0 \le y \le W$) to pinpoint maximum hotspot temperatures at interior cores or boundary heat pipe interfaces, preventing thermal throttling or silicon delamination.
• Structural Engineering: Bending stress $\sigma(x, y)$ across structural beam cross-sections subjected to combined axial load and biaxial bending must be evaluated across the perimeter of the cross-section to verify yield safety margins.

Example Problems

Example 1: Absolute Extrema on a Closed Rectangular Region

Find the absolute maximum and minimum values of:

Example 2: Absolute Extrema on a Triangular Region

Find the absolute extrema of $f(x, y) = x^2 + y^2$ on the closed triangular region bounded by the vertices $(0, 0)$, $(4, 0)$, and $(0, 3)$.

Connections & References

• Prerequisites:
  • Critical Points in Multivariable Functions — Finding interior stationary points.
  • Partial Derivatives — Differentiating multi-input functions.
• Direct Successors:
  • Lagrange Multipliers — Systematic method for finding extrema along curved or implicit boundaries.
• Course Sessions:
  • Session 13 - Optimization with Constraints