Study polynomial critical points with confidence. Check maxima, minima, inflection hints, and derivative sign changes. Plot curves, inspect coordinates, export tables, and save reports.
This calculator analyzes the quartic model f(x) = ax^4 + bx^3 + cx^2 + dx + e.
You may set leading coefficients to zero to analyze cubic, quadratic, linear, or constant cases.
| Example Function | Derivative | Critical x-values | Point Types | Point Coordinates |
|---|---|---|---|---|
| f(x) = x^4 - 4x^2 + 3 | f'(x) = 4x^3 - 8x | -1.414214, 0, 1.414214 | Min, Max, Min | (-1.414214, -1), (0, 3), (1.414214, -1) |
These values come from the default example already loaded into the form.
f(x) = ax^4 + bx^3 + cx^2 + dx + e.
f'(x) = 4ax^3 + 3bx^2 + 2cx + d.
f'(x) = 0.
Real solutions are the candidate critical points.
f''(x) = 12ax^2 + 6bx + 2c.
If f''(x) > 0, the point is a local minimum.
If f''(x) < 0, the point is a local maximum.
A critical point is an x-value where the first derivative is zero or undefined, while the point still belongs to the function’s domain.
No. Some critical points are stationary inflection points. The calculator checks the second derivative and nearby derivative signs to classify them better.
Yes. Set the higher-degree coefficients to zero. The calculator automatically reduces the model and solves the correct derivative equation.
If the derivative equation has no real roots, then the function has no real critical points. This often happens when the graph never flattens.
It measures curvature. Positive values suggest a local minimum, while negative values suggest a local maximum near the tested critical point.
They show where the function is increasing or decreasing. This helps confirm whether each critical point behaves like a peak, valley, or neither.
Its derivative is zero everywhere, so every real x-value is critical. However, there are no isolated turning points because the output never changes.
They include the function, derivatives, coefficient values, summary text, critical point data, and interval behavior from the current calculation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.