Peak Finder: Where the Slope Is Zero, Peak or Valley, and the Ends

Opens on Chapter 12’s example: the fair ride’s track, x³ − 12x + 20, from x = −3 to 3.

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How it works

The slope of a smooth curve is zero at the top of a hill and the bottom of a valley, so solve slope = 0 (the critical points). The sign test: if the slope is + before and − after, it's a peak; − then +, a valley; the same sign on both sides, neither (a flat spot on the way up or down). The bend test does the same job with the second derivative: negative (frowning) at a peak, positive (cupping) at a valley. On an interval, the overall highest or lowest point can be at an end, so compare the critical points with both ends.

From the book, Chapter 12. The throw peaks at t = 1.25 s, 30 ft. The day warms to 85°F at t = 10 (4 p.m.); between dawn and noon the warmest moment is the end, 74.4°F, still rising 4.3°F an hour. The attic is cheapest at 6 inches, $240 a year. The ride crests at 36 ft and dips to 4, but from −3 to 4.5 the end is highest, 57.1 ft. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.