Global Extrema

Calculus 3 · Axiom Academy

Finding the absolute maximum and minimum of a continuous function on a closed, bounded region — interior critical points plus the whole boundary. Before we hunt for extrema, we need to know they are actually there. The Extreme Value Theorem is that guarantee — no matter how complicated f is, on the right kind of region it must reach a highest value and a lowest value. The theorem promises the max and min exist — but where ? There are exactly two possibilities: a point in the interior where the surface goes flat, or a point on the boundary edge. Our whole strategy is to round up every candidate from both places. Solve for critical points strictly inside D . Parametrize the edge (or use Lagrange multipliers) and optimize there. Evaluate f at every candidate; largest is the max, smallest the min. Unlike the single-variable case — where the "boundary" is just two endpoints — the boundary in 2D is a whole curve that must be examined along its entire length. 3. Step 1 — Interior Critical Points Inside the region, an extremum can only happen where the tangent plane is horizontal — that is, where the gradient vanishes. So the first candidates are the solutions of that lie strictly inside D . Take f(x,y) = x^2 + y^2 - 2x - 4y on the disk . Check it is inside: ✓︎. And f(1,2) = 1 + 4 - 2 - 8 = -5. 4. Step 2 — Scanning the Boundary

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