Before you can run the recipe in the parent topic, you must be able to read it. Below is every symbol and idea the parent uses, in an order where each one leans only on the ones before it.
Why the topic needs it: the parent's goal — "find the absolute max and min" — is meaningless until we agree these mean the outright tallest and shortest points over all of D.
Figure s01 — Alt text: A curved surface (terrain) floats above a flat x–y floor. One floor address (x,y) is marked with a navy dot; a vertical navy line rises from it to a magenta dot on the surface labelled f(x,y), showing that the function value is the height above that address.
The floor is the flat plane of all addresses (x,y).
Above each address sits a dot at height f(x,y).
All those dots together form a surface — the "terrain".
Why the topic needs it: the whole game is "find the top and bottom of this terrain", so we first need to know that a terrain is what f draws.
Figure s02 — Alt text: A right triangle inside the unit circle. The horizontal leg (length x) and vertical leg (length y) meet at a right angle; the dashed hypotenuse from the origin to the point (x,y) is labelled r=x2+y2, illustrating Pythagoras.
x2+y2<1, i.e. r<1 → strictly inside the circle of radius 1 (the open disk).
x2+y2=1, i.e. r=1 → exactly ON the circle (the boundary).
x2+y2≤1, i.e. r≤1 → inside OR on it (the closed disk).
Why the topic needs it: the disk region and the "is (21,21) inside?" check both come straight from comparing r2=x2+y2 to 1. In Example 2, 41+41=21<1, so r<1 and the point is safely inside.
Figure s03 — Alt text: A shaded disk with a solid violet boundary circle. A magenta dot near the middle is labelled "interior point (wiggle room all ways)"; a navy dot on the boundary circle is labelled "boundary point (on the fence)". A dotted larger circle around everything is labelled "fits in a big circle → bounded".
Look at the figure: a point in the pale middle is interior; a point on the dark rim is boundary. If the rim is drawn solid, the set owns it — that's closed.
Why the topic needs it: the parent's strategy literally splits into "interior" vs "boundary", so these words are the coat-hooks the whole recipe hangs on.
Figure s05 — Alt text: Two side-by-side height slices. On the left, a smooth continuous curve labelled "continuous — small step in x → small change in height". On the right, a curve with a vertical jump (a torn cliff) at one point, an open circle below and a filled circle above, labelled "discontinuous — a tiny step causes a sudden jump"; a magenta arrow points at the gap.
Why the topic needs it: if the sheet had a rip, the "highest point" could hide at the torn edge and never be attained. Continuity is a required ingredient of the Extreme Value Theorem.
Figure s04 — Alt text: A height-versus-step graph showing two curves — an orange parabola (the slice with y fixed, whose slope is fx) and a magenta line (the slice with x fixed, whose slope is fy). A dashed violet tangent line touches the orange curve at a marked point, labelled "steepness here = fx".
Why derivatives and not something else? A derivative answers exactly the question "which way is uphill, and how steep?" — the precise question we ask to find a flat spot. This machinery is developed in Critical points and gradient.
Figure s06 — Alt text: A surface with a contour/floor beneath it. At one floor point a violet arrow labelled "∇f" points in the steepest-uphill direction; a magenta dashed path shows the surface climbing most steeply above that arrow. At a separate flat dimple the arrow has shrunk to a dot labelled "∇f=0 — level, a critical point".
Why the topic needs it: Step 1 of the recipe is literally "solve ∇f=0 inside D".
Why the topic needs it: every boundary piece — every edge of a rectangle, the whole rim of a disk, and every joining corner — must be squeezed into a form where ordinary 1D methods apply.
The diagram below is a dependency chart: an arrow "A → B" means "you need A before B makes sense". Read it top-to-bottom. The three raw ingredients — the terrain f, Pythagoras (for distance), and continuity — feed into the shape-words and derivatives; those in turn power the two recipe steps, which finally combine into the parent recipe. The figure just below renders the same structure as a picture so you can see the flow, not just read it.
Figure s07 — Alt text: A top-down dependency flowchart. Boxes for "function f", "Pythagoras (x²+y²)", and "continuous" sit at the top; arrows lead down through "region D", "interior/boundary/closed/bounded", "partial derivatives", and "gradient ∇f" into two boxes "Step 1 interior critical points" and "Step 2 boundary as one variable", which both feed the bottom box "Absolute extrema recipe". "Extreme Value Theorem 2D" also feeds the recipe.
What makes a point a global maximum (not just "high")?
Its height is ≥ the height at every other point of D — tallest over the WHOLE field, not merely taller than nearby bumps.
What does f(x,y) represent as a picture?
A height/terrain floating above each floor address (x,y).
What does {0≤x≤3,0≤y≤2} describe?
The set of all points with x between 0 and 3 AND y between 0 and 2 — a rectangle.
What is r as a formula, and why is it never negative?
r=x2+y2, the distance from the origin; a distance is a length, so the square root takes its non-negative value.
Why does x2+y2 work even when x or y is negative?
Squaring erases the sign, so it gives the correct distance-squared in all four quadrants.
Difference between x2+y2<1, =1, and ≤1?
r<1 strictly inside; r=1 exactly on the boundary; r≤1 inside or on it (closed disk).
What do "closed" and "bounded" each mean?
Closed = includes its boundary fence (≤); bounded = fits inside some finite circle.
Why must f be continuous for the guarantee?
A rip lets height approach a value without ever reaching it, so no genuine highest point exists.
What is fx and what is held fixed?
The slope walking in the x-direction, holding y constant.
What is ∇f and what does ∇f=0 mean geometrically?
The arrow (fx,fy) pointing steepest uphill; zero means the ground is level in every direction (a critical point).
Why can an interior max only occur where ∇f=0?
If the arrow were nonzero you could step uphill and go higher, so it wasn't the max.
How do you handle all four edges and corners of a rectangle boundary?
Fix the pinned coordinate on each edge (making f a 1-variable function), find its interior critical points, and always include the four corner endpoints as candidates.
How do you turn the unit circle boundary into one variable?
Set x=cost,y=sint and let the angle t be the single dial.
