Critical Numbers Calculator
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: December 15, 2024
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Use x² for squared, x³ for cubed, etc.
Critical Numbers Analysis
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
Step-by-Step Solution:
Analysis:
The cubic function has two critical numbers where the derivative equals zero
Tips for Finding Critical Numbers:
- • Critical numbers occur where f'(x) = 0 or f'(x) is undefined
- • Always verify critical numbers are in the domain of the original function
- • Use the first derivative test to classify each critical number
- • For optimization problems, check endpoints and critical numbers
- • Second derivative test: f''(c) > 0 means minimum, f''(c) < 0 means maximum
Critical Numbers Applications & Types
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
Quick Example Result
For function f(x) = x³ - 3x² + 2:
Critical Number 1
x = 0
Local Maximum
Critical Number 2
x = 2
Local Minimum
How to Find Critical Numbers
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.
The Critical Numbers Process
This systematic approach ensures all critical numbers are identified and properly classified.
First Derivative Test
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.
- If f'(x) changes from + to -, the critical number is a local maximum
- If f'(x) changes from - to +, the critical number is a local minimum
- If f'(x) doesn't change sign, it's neither (saddle point or inflection)
- The second derivative test can also classify critical numbers
Sources & References
- Calculus: Early Transcendentals - James Stewart (9th Edition)Comprehensive coverage of critical numbers and extrema
- Thomas' Calculus - Weir, Hass, Giordano (14th Edition)Detailed explanations of derivative tests and applications
- Wolfram Alpha - Critical Points and ExtremaComputational tool for verifying critical number calculations
Need help with other calculus topics? Check out our derivative calculator and concavity calculator.
Get Custom Calculator for Your PlatformCritical Numbers Example
Given Function:
Solution Steps:
- Step 1: Find the first derivative f'(x)
- f'(x) = 3x² - 6x
- Step 2: Set f'(x) = 0 and solve
- 3x² - 6x = 0
- 3x(x - 2) = 0
- x = 0 or x = 2
- Step 3: Check for undefined points
- No undefined points in the derivative
Critical Numbers: x = 0, 2
At x = 0:
Local Maximum
f(0) = 2
At x = 2:
Local Minimum
f(2) = -2
Quadratic Example
f(x) = x² - 4x + 3
Critical number: x = 2 (minimum)
Rational Example
f(x) = x + 1/x
Critical numbers: x = ±1
Frequently Asked Questions
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