Critical Number
Critical
Number
Critical Value
The x-value of a critical point.
Worked Example
Problem: Find the critical numbers of f(x) = x³ − 6x² + 9x + 1.
Step 1: Take the derivative of f(x).
f′(x)=3x2−12x+9
Step 2: Set the derivative equal to zero and solve.
3x2−12x+9=0⟹3(x2−4x+3)=0⟹3(x−1)(x−3)=0
Step 3: Identify the solutions. Since f'(x) is a polynomial, it is defined everywhere, so the only critical numbers come from setting f'(x) = 0.
x=1andx=3
Answer: The critical numbers are x = 1 and x = 3.
Why It Matters
Critical numbers are the starting point for finding where a function reaches its highest or lowest values. The First Derivative Test and Second Derivative Test both rely on critical numbers to classify local extrema. When solving optimization problems in calculus, you identify critical numbers first, then determine which one gives the optimal solution.
Common Mistakes
Mistake: Forgetting to check where the derivative is undefined, not just where it equals zero.
Correction: A critical number occurs at any x in the domain of f where f'(x) = 0 OR where f'(x) does not exist. For example, f(x) = |x| has a critical number at x = 0 because f'(0) is undefined, even though f'(x) never equals zero at that point.
Related Terms
- Critical Point — The point (x, f(x)) at a critical number
- First Derivative Test — Uses critical numbers to classify extrema
- Local Maximum — A type of extremum found at critical numbers
- Local Minimum — A type of extremum found at critical numbers
- Derivative — Must be zero or undefined at a critical number