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Free critical numbers calculator for finding critical points using derivatives. Get step-by-step solutions with the first derivative test and extrema analysis. Perfect for calculus students learning to identify local maxima, minima, and optimization problems.
Last updated: February 2, 2026
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Use x² for squared, x³ for cubed, etc.
First Derivative:
f'(x) = 3x² - 6x
Critical Numbers Found:
x = 0
x = 2
Extrema Classification:
At x = 0
Local Maximum
f(0) = 2
At x = 2
Local Minimum
f(2) = -2
Analysis:
The cubic function has two critical numbers where the derivative equals zero
Test condition
f'(x): + → 0 → -
Derivative changes from positive to negative at critical point
Test condition
f'(x): - → 0 → +
Derivative changes from negative to positive at critical point
Test condition
No sign change
Derivative doesn't change sign at critical point
Example
f(x) = x³ - 3x²
Derivative always defined, solve f'(x) = 0 for critical numbers
Consider
f'(x) = 0 and undefined
Check both where derivative equals zero and is undefined
Applications
Max Profit, Min Cost
Find optimal solutions by analyzing critical numbers
For function f(x) = x³ - 3x² + 2:
Critical Number 1
x = 0
Local Maximum
Critical Number 2
x = 2
Local Minimum
Finding critical numbers is a fundamental skill in calculus that helps identify potential extrema and understand function behavior. The process involves derivative analysis and systematic evaluation of where the rate of change equals zero or becomes undefined.
This systematic approach ensures all critical numbers are identified and properly classified.
The first derivative test determines whether a critical number is a local maximum, minimum, or neither by examining the sign of the derivative before and after the critical point. This test is essential for optimization problems and understanding function behavior.
Need help with other calculus topics? Check out our derivative calculator and concavity calculator.
Get Custom Calculator for Your PlatformCritical Numbers: x = 0, 2
At x = 0:
Local Maximum
f(0) = 2
At x = 2:
Local Minimum
f(2) = -2
f(x) = x² - 4x + 3
Critical number: x = 2 (minimum)
f(x) = x + 1/x
Critical numbers: x = ±1
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