Critical Points Calculator

Critical points are x-values where the derivative of a function equals zero or is undefined. For a smooth polynomial, it is always zero — those are the spots where the function momentarily flattens out before continuing up, down, or reversing direction.

This calculator works with cubic polynomials: f(x) = ax³ + bx² + cx + d. The first derivative is f’(x) = 3ax² + 2bx + c, which is a quadratic. Setting that equal to zero and applying the quadratic formula gives the critical points (if any exist).

Discriminant: D = (2b)² - 4·(3a)·c = 4b² - 12ac

If D > 0: two critical points at x = (-2b ± √D) / (6a) If D = 0: one critical point (the parabola touches zero at exactly one place) If D < 0: no real critical points (the derivative never reaches zero)

Once you have a critical point x₀, the second derivative f’’(x) = 6ax + 2b tells you its nature:

• f’’(x₀) > 0: local minimum (curve is concave up)
• f’’(x₀) < 0: local maximum (curve is concave down)
• f’’(x₀) = 0: inconclusive: could be a saddle point or inflection point

The chart plots f(x) over a range centered on the critical points so you can see the curve shape directly. A local max looks like a hill; a local min looks like a valley.

Cubic polynomials have zero, one, or two critical points, and the chart marks whichever ones exist in red.

Set a = 0 and the function drops to a quadratic, which behaves differently: it has exactly one critical point, its vertex, and that one is a genuine global extremum. Set b to zero as well and you are left with a straight line, which has none at all, unless c is also zero and the whole thing is a flat constant, in which case every point is critical and none of them means anything.

One thing worth keeping straight: on a real cubic, a critical point is only ever a local extremum. The tails run to +∞ and −∞, so there is no global maximum or minimum to find. A hill on a cubic is a hill on the way up, not the top of the mountain.