The curve f of x equals x to the fourth over four minus x squared, with two valleys at plus and minus root two and a hilltop at zero. A draggable point carries the tangent line; readouts give f, the slope f prime equals x cubed minus two x, the second derivative three x squared minus two, and a verdict: rising, falling, or flat as a valley or hilltop. A step-size slider and buttons take gradient-descent steps, x moves to x minus the step size times the slope, leaving a trail of dots; the steps shrink as the slope flattens and land in the nearest valley.

Read the slope, then step downhill

f(x) = x⁴/4 − x². Drag the point and read f′ and f″; then take steps against the slope and watch where you land.