Critical Points Calculator (Cubic)
Find critical points of a cubic and classify them.
Enter your values
Enter the coefficient of x^3 you are working with.
Enter the coefficient of x^2 you are working with.
Enter the coefficient of x you are working with.
Enter the constant you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- Critical point 1
- x = 2 (local minimum)
- Critical point 2
- x = 0 (local maximum)
- f'(x)
- 3x^2 + -6x + 0
Quick answer
Critical points occur where f'(x)=0; the second derivative test classifies them as local maxima, minima or inconclusive.
f'(x)=3ax^2+2bx+c=0
At a glance
| What it does | Find critical points of a cubic and classify them. |
|---|---|
| Category | Math |
| Inputs needed | Coefficient of x^3, Coefficient of x^2, Coefficient of x, Constant |
| Main output | Critical point 1 |
| Formula | f'(x)=3ax^2+2bx+c=0 |
| Cost | Free — no sign-up, no download |
How to use the Critical Points Calculator (Cubic)
- 1Enter the coefficients of a cubic polynomial.
- 2The derivative is set to zero and solved with the quadratic formula.
- 3The second derivative test classifies each critical point.
Inputs explained
- Coefficient of x^3
- Enter the coefficient of x^3 you are working with.
- Coefficient of x^2
- Enter the coefficient of x^2 you are working with.
- Coefficient of x
- Enter the coefficient of x you are working with.
- Constant
- Enter the constant you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- Coefficient of x^31
- Coefficient of x^2-3
- Coefficient of x0
- Constant2
Results
- Critical point 1x = 2 (local minimum)
- Critical point 2x = 0 (local maximum)
- f'(x)3x^2 + -6x + 0
Frequently asked questions
What if the discriminant is negative?
The derivative never crosses zero, so the cubic is monotonic with no local extrema.
What does 'inconclusive' mean?
The second derivative is also zero there, so a higher-order test or sign chart is needed.
What do I need to enter into the Critical Points Calculator (Cubic)?
Just 4 values: coefficient of x^3, coefficient of x^2, coefficient of x, constant. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Critical Points Calculator (Cubic) work out the answer?
Critical points occur where f'(x)=0; the second derivative test classifies them as local maxima, minima or inconclusive. It applies the formula f'(x)=3ax^2+2bx+c=0 and shows the working so you can check each step by hand.
Is the Critical Points Calculator (Cubic) free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.