Critical Points Calculator
Find and classify critical points of a cubic polynomial ax³ + bx² + cx + d. Use it when you need to locate local maxima, minima, and inflection points quickly.
About this calculator
For a cubic polynomial f(x) = ax³ + bx² + cx + d, critical points occur where the first derivative equals zero: f′(x) = 3ax² + 2bx + c = 0. Solving this quadratic gives the x-coordinates of potential maxima and minima. To classify each critical point, evaluate the second derivative f″(x) = 6ax + 2b at the candidate point. If f″(x) > 0 the point is a local minimum; if f″(x) < 0 it is a local maximum; if f″(x) = 0 the second derivative test is inconclusive and further analysis is needed. This calculator computes f″(test_point) = 6a·x + 2b to assist with that classification step.
How to use
Consider f(x) = x³ − 3x² − 9x + 5 (a = 1, b = −3, c = −9, d = 5). Step 1: Set f′(x) = 3x² − 6x − 9 = 0, giving x = −1 and x = 3. Step 2: To classify x = 3, enter test_point = 3. Step 3: f″(3) = 6·(1)·3 + 2·(−3) = 18 − 6 = 12 > 0, so x = 3 is a local minimum. Step 4: Try test_point = −1: f″(−1) = −6 − 6 = −12 < 0, so x = −1 is a local maximum.
Frequently asked questions
How do I find critical points of a cubic function step by step?
Differentiate f(x) = ax³ + bx² + cx + d to get f′(x) = 3ax² + 2bx + c. Set f′(x) = 0 and solve the resulting quadratic using the quadratic formula or factoring. The solutions are the x-values of the critical points. Substitute each back into f(x) to find the corresponding y-values, giving you the coordinates of each critical point.
What does the second derivative tell you about a critical point?
The second derivative test determines whether a critical point is a local maximum, minimum, or neither. If f″(x) > 0 at the critical point, the curve is concave up there, indicating a local minimum. If f″(x) < 0, the curve is concave down, indicating a local maximum. If f″(x) = 0, the test is inconclusive and you must inspect the sign of f′ on either side of the point.
Can a cubic polynomial have no critical points?
Yes. A cubic polynomial has critical points only if its first derivative — a quadratic — has real roots. This requires the discriminant 4b² − 12ac ≥ 0. If the discriminant is negative, the derivative is always positive or always negative, meaning the cubic is strictly monotone with no local extrema. In that case the function still has an inflection point, but no maximum or minimum.