A critical number calculator finds the critical numbers (or critical points) of a function, the x-values where the derivative is zero or undefined. These points are where a function can have local maximums, local minimums or changes in direction, which makes them central to calculus and optimization problems.
The steps: find the derivative f′(x), set it equal to zero and solve, then find where f′(x) is undefined while f(x) is defined. Example: f(x) = x³ − 6x² + 9x + 1. The derivative is f′(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3). Setting it to zero gives critical numbers x = 1 and x = 3. Plugging in, f(1) = 5 and f(3) = 1, giving the critical points (1, 5) and (3, 1).
To classify them, use the first derivative test (check the sign of f′ on either side) or the second derivative test. Here, f″(x) = 6x − 12. At x = 1, f″ = −6 (concave down), so it’s a local maximum. At x = 3, f″ = 6 (concave up), so it’s a local minimum. If f″ = 0, the test is inconclusive.
Common mistakes: forgetting points where the derivative is undefined, like corners or cusps (f(x) = |x| has a critical number at 0), and including points outside the function’s domain. For absolute extremes on a closed interval, also check the endpoints.
Why use an online Critical Number Calculator: it saves time, avoids simple arithmetic mistakes and shows each step, so you can check your homework, plan a project or make a decision with confidence. Bookmark the page so it’s ready the next time you need it.
Tips for getting accurate results from the Critical Number Calculator: double-check that every input uses the right units, enter numbers without commas or symbols, and run the calculation a second time with a quick estimate in your head to confirm the answer looks reasonable.
Learn more on Wikipedia’s critical point article. Find concavity, implicit differentiation and limit calculators on Minedit.