Step 1: Group the costs into "totals" and "per-unit averages":
Totals: TFC (flat), TVC (inverse-S, rises with output), TC = TFC+TVC (same shape as TVC, shifted up). Per-unit: AFC = TFC/Q, AVC = TVC/Q, AC = TC/Q = AFC+AVC. Rate of change: MC = ΔTC/ΔQ.
Step 2: Explain the U-shape intuition via the law of variable proportions:
As a fixed factor (e.g. a factory) is combined with more and more of a variable factor (labour), output per worker first rises (falling AVC/MC) due to specialisation, then falls (rising AVC/MC) due to overcrowding/diminishing returns — this single law explains why AVC, AC and MC are all U-shaped.
Step 3: Prove the MC-minimum-AC intersection with simple logic (no calculus needed):
If the cost of the next unit (MC) is less than the current average, adding it must pull the average down; if MC exceeds the average, adding it pulls the average up. The average can only stop falling and start rising exactly at the point where MC crosses it — hence MC must intersect AC (and AVC) precisely at their minimum points.
Step 4: Diagram description:
Sketch AFC as a smoothly falling curve hugging the axes; AVC and AC as U-shaped curves with AC always above AVC by a shrinking gap (=AFC); MC as a steeper U-shaped curve piercing both AVC and AC exactly at their lowest points.
Final Answer:
All the short-run per-unit cost curves (AFC falling, AVC/AC/MC U-shaped) derive from the fixed-variable cost split and the law of variable proportions, with MC always crossing AVC and AC at their minimum points.