Critical Numbers Calculator
Find the critical numbers of any function. Enter f(x) and get every point where the derivative equals zero or does not exist, classified as local max, local min or inconclusive using the second derivative test.
Cubic with two critical points
Critical Numbers Found: 2
| # | x (critical number) | f(x) | f'(x) | f''(x) | Classification |
|---|---|---|---|---|---|
| 1 | -1.000000 | 3.000000 | 0 | -6.0001 | local maximum |
| 2 | 1.010000 | -0.999699 | 0 | 6.0603 | local minimum |
Solution Steps
1. Given: f(x) = x^3 - 3*x + 1
2. Find f'(x) and set f'(x) = 0 (numerical approach used)
3. Also check where f'(x) does not exist (cusps, corners, vertical tangents)
4. Found 2 critical numbers in [-10, 10]
5. Second derivative test applied: f''(c) > 0 means local min, f''(c) < 0 means local max
Summary
Critical Numbers
2
f(x) = 3.0000
f(x) = -0.9997
Second Derivative Test
Automatic Detection
Scans the entire range for sign changes in f'(x) and points where the derivative does not exist. Finds all critical numbers, not just obvious ones.
Second Derivative Test
Classifies each critical point as local maximum, local minimum or inconclusive by evaluating f''(x) at the critical number.
Step-by-Step
Full solution walkthrough showing the function, derivative analysis, critical numbers found and classification reasoning.
How to Find Critical Numbers Step by Step
Critical Numbers in Calculus: Everything You Need to Know
Definition and Importance
A critical number of a function f is a number c in the domain of f such that either f'(c) = 0 or f'(c) does not exist. Critical numbers are the backbone of curve sketching and optimization in calculus. Every local maximum and local minimum of a continuous function must occur at a critical number (this is Fermat's theorem). If you want to find where a function reaches its highest or lowest values, critical numbers are where you look.
Finding Critical Numbers by Hand
The process is straightforward for polynomial and rational functions:
- Compute f'(x) using differentiation rules.
- Set the numerator of f'(x) equal to zero and solve for x. These give f'(x) = 0.
- Find where the denominator of f'(x) equals zero (if f'(x) is a fraction). These are points where f'(x) does not exist.
- Keep only the values that are in the domain of the original function f.
For f(x) = x^3 - 3x + 1, the derivative is f'(x) = 3x^2 - 3 = 3(x-1)(x+1). Setting this to zero gives x = 1 and x = -1. Both are in the domain, so both are critical numbers.
Classifying Critical Points
Finding critical numbers is step one. Step two is determining what happens at each one. There are two common tests:
First derivative test: Check the sign of f'(x) on either side of the critical number. If f' changes from positive to negative, it is a local maximum. If f' changes from negative to positive, it is a local minimum. If f' does not change sign, it is neither (like x = 0 for x^3).
Second derivative test: If f'(c) = 0 and f''(c) exists, check the sign of f''(c). Positive means the graph is concave up (a bowl shape), so c is a local minimum. Negative means concave down (an arch), so c is a local maximum. Zero means the test gives no information and you must fall back to the first derivative test.
Types of Critical Numbers
- Where f'(c) = 0: The tangent line is horizontal. These are the most common type and include smooth peaks, valleys and inflection points with horizontal tangents.
- Cusps: The function is continuous but the derivative blows up. Example: f(x) = x^(2/3) at x = 0. The graph comes to a sharp point.
- Corners: The left and right derivatives exist but differ. Example: f(x) = |x| at x = 0. The graph has a V-shape.
- Vertical tangents: The derivative approaches infinity. Example: f(x) = x^(1/3) at x = 0.
Absolute vs Local Extrema
On a closed interval [a, b], the extreme value theorem guarantees that a continuous function attains an absolute maximum and minimum. To find them, evaluate f at every critical number inside (a, b) and also at the endpoints a and b. The largest value is the absolute maximum and the smallest is the absolute minimum. This is the closed interval method and it is the standard approach for optimization problems on bounded domains.
Common Mistakes
- Forgetting to check where f'(x) is undefined, not just where it equals zero.
- Including values outside the domain of f. If f(x) = sqrt(x), then x = -1 is not a critical number even if some algebra step produces it.
- Assuming every critical number is a max or min. Inflection points with horizontal tangents (like x^3 at x = 0) are critical numbers but not extrema.
- Forgetting endpoints on closed intervals. The absolute max or min might be at an endpoint, not at a critical number.