Enter a polynomial up to degree 3 and instantly find its critical points, classified as local maxima, local minima, or inconclusive.
Maxima & Minima Calculator
f(x) = a·x³ + b·x² + c·x + d
How It’s Calculated
Critical points occur where f'(x) = 0. For a cubic f(x) = ax³+bx²+cx+d, that means solving the quadratic 3ax²+2bx+c = 0. At each critical point, the second derivative test classifies it: f”(x) > 0 means the curve is concave up there (a local minimum), f”(x) < 0 means concave down (a local maximum), and f”(x) = 0 is inconclusive from this test alone — it needs a first-derivative sign check on either side instead.
Worked Example
For f(x) = x³ − 3x²: f'(x) = 3x² − 6x = 3x(x−2), giving critical points x = 0 and x = 2. f”(x) = 6x−6. At x=0: f”=−6<0, local maximum, f(0)=0. At x=2: f”=6>0, local minimum, f(2)=−4.
Frequently Asked Questions
Q: What if the second derivative test is inconclusive?
Check the sign of f'(x) just before and just after the critical point by hand. If it changes from positive to negative, it’s a maximum; negative to positive, a minimum; no change, it’s neither (an inflection point).
Q: Does this work for degree 4 and higher?
Not this tool — it’s scoped to degree 3, since the derivative is then a quadratic solvable by formula. Higher-degree critical-point equations generally don’t have a clean closed-form solution.
This is exactly the Application of Derivatives technique tested throughout 2nd PUC. See how Daniel Classes teaches 1st and 2nd PUC Mathematics →