Peak and Valley Finder

Calculus 1 · Axiom Academy

At the very top of a hill or the bottom of a valley, the ground goes flat. That single fact — the slope is zero — is how the derivative hands you every peak and valley on a curve. The flat spot at every top and bottom Imagine hiking over a hill. Going up, the trail tilts up; coming down, it tilts down — and for one instant at the very top, you walk on flat ground. The slope of the path is exactly zero there. A valley is the same idea upside down. The derivative measures that slope, so the peaks and valleys of a curve are exactly the spots where the derivative is zero. Watch the sweep line carry a little tangent along the curve. Keep your eye on its tilt: it leans uphill, goes flat at the peak, then leans downhill — and the sign of the slope + → 0 → − tells you a peak just passed. At the valley the pattern reverses: − → 0 → + . A peak is where the slope hands off from + to − ; a valley, from − to + . Both pass through slope = 0. Slide the point and watch the slope flip Here is a real curve, f(x) = x^3 - 3x . Drag the point along it. The tangent line and the live slope f'(x) change as you move — uphill the slope is positive, downhill it is negative, and it reads exactly zero right as you cross a peak or a valley. Hunt for the two spots where f'(x) = 0 . Two flat spots: x=-1 and x=+1 . Those are the candidates — a peak or a valley sits at each. Which is which? That is the next move. Peak or valley? Read the signs on either side

This is the written version of the interactive lesson above. See the full Calculus 1 course.