How to Use the Increasing/Decreasing Interval Calculator
For a cubic function y=ax^3+bx^2+cx+d, the increasing and decreasing intervals are determined by the sign of its derivative, y'=3ax^2+2bx+c. Where the derivative is positive, the original function is increasing; where it's negative, the function is decreasing. This calculator takes coefficients a, b, c, and d, finds the critical points (where the derivative equals zero) using the derivative's discriminant, and analyzes the sign on either side to report the increasing/decreasing intervals along with the local maximum and minimum.
When the derivative's discriminant D=4b^2-12ac is less than 0, there are no critical points, so the function keeps increasing or keeps decreasing across the entire domain (always increasing if a is positive, always decreasing if a is negative). When it's greater than 0, there are two critical points, and depending on the sign of a, the function follows an increase-decrease-increase or decrease-increase-decrease pattern, with a local max or min at each critical point.
The coefficient a must be nonzero for the function to actually be cubic — if a is 0, it reduces to a quadratic or lower, and this calculator's method no longer applies.
Frequently Asked Questions
Usually yes when a is positive — the earlier critical point gives the local max and the later one gives the local min, with the max being the larger value. When a is negative, the order is reversed.
When the discriminant is exactly 0, the derivative touches zero at that point but doesn't change sign, so it behaves more like an inflection point than a true local extremum. This calculator reports that case as the function being monotonically increasing or decreasing across the whole domain.