Practice question · Multiple choice
A farmer with 100 m of fence wants the largest rectangular field. Calculus says a square. Why does the derivative find a maximum here without checking every shape?
Hints
- Sketch area against width. What is the slope at the peak?
- Ask how many rectangles you would otherwise have to check.
Show the answer
D. Because at the top of a smooth curve the tangent is horizontal
Why
Area against width is a downward parabola, and its peak has zero slope, so one equation replaces an infinite search. That is what makes calculus an engineering tool rather than a curiosity: 'find the best' becomes 'solve f'(x) = 0', which is the same move behind every least-squares fit and every training run.
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