A cubic function
f(x) = x³ − 3x + 2 has zeros at x = 1 (a repeated root) and x = −2, a local maximum at x = −1 and a local minimum at x = 1, and an inflection point at x = 0.
Polynomial function study
Result
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How it works
A polynomial's domain is always every real number, so the study moves straight to more interesting questions: where the function crosses zero, where its slope is zero (the derivative's own zeros, marking peaks, valleys or saddle points), and where its curvature flips from concave to convex or back (the second derivative's zeros, the inflection points).
The derivative of a polynomial is computed symbolically term by term (the power rule), which is exact — no approximation. Zeros of the function, the derivative and the second derivative are then found by scanning for sign changes and refining each one, since there is no general formula for roots of a polynomial above degree 4.
Formulas used
Power rule
d/dx(xⁿ) = n·xⁿ⁻¹, applied term by term
Worked examples
A simple parabola
f(x) = x² − 4 has zeros at x = −2 and x = 2, and a single critical point at x = 0, a local minimum.
Assumptions and limits
- The function is a polynomial with real coefficients, degree 6 or lower.
- Roots are searched for within ±50 — a function with roots outside that range may show an incomplete picture.
Frequently asked questions
How do I enter the coefficients?
List them from the highest power of x down to the constant term, separated by commas — for f(x) = 2x³ − 5x + 1, enter "2, 0, -5, 1" (note the 0 for the missing x² term).
What is a critical point?
A point where the derivative equals zero — the function's slope is momentarily flat there, marking a local maximum, a local minimum, or (rarely, for a polynomial) a saddle point where the slope touches zero but does not actually turn around.
What is an inflection point?
A point where the second derivative equals zero and the curve switches from curving upward (convex) to curving downward (concave), or vice versa — the graph's "S-bend" points.
Why does this only handle polynomials?
Polynomials have exact, symbolic derivatives that are simple to compute term by term. Rational, trigonometric or exponential functions need a more general symbolic engine, which is out of scope for this tool.
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