Where slopes vanishmaxima and minima
At the very top of a hill the tangent goes flat. Set the derivative to zero and you can find the best of anything.
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Toss a ball straight up and plot its height against time. Somewhere on that arc is the moment it peaks, and eyeballing gets you close, but close isn't an answer. We want the precise instant, to the decimal.
Stop asking about height and ask about slope. On the way up, the ball is gaining — slope positive. Past the peak, it's losing — slope negative. Every point on the curve tilts one way or the other. Almost every point.
Between positive and negative there has to be a crossover, and that crossover is the summit. Right at the top, the tangent lies flat — slope zero. For one instant the ball is neither rising nor falling, and that instant is the peak.
That turns a guessing game into algebra. Height is twenty t minus five t squared. Its derivative: twenty minus ten t. Set that to zero and solve — t equals two seconds. Back into the original: twenty meters. The exact top of the arc.
Valleys follow the same rule. At the bottom of a U, the tangent flattens — slope zero again. The only difference is the order: negative, then zero, then positive. Falling, flat, rising. One equation finds the lowest point the way it finds the highest.
So the guess that got us close became an exact answer, and that same move has a name: optimization. The cheapest route, the strongest beam, the best price, the training run that finishes fastest — each is the top or bottom of some curve, and each is one flat tangent away. Best means the slope is zero. Take the derivative, set it to zero, solve — that's how you find the peak of anything.