Critical Points Calculator

About the Critical Points Calculator

The Critical Points Calculator helps you find the critical points of a function, where the derivative is zero (\( f'(x) = 0 \)) or undefined. It supports functions like \( x^3 - 3x + 1 \), \( \sin(x) \), or \( e^x \). This tool is ideal for calculus students, teachers, and professionals.

• Features:
  • Computes the derivative of the input function (e.g., \( f(x) = x^3 - 3x + 1 \rightarrow f'(x) = 3x^2 - 3 \)).
  • Finds points where \( f'(x) = 0 \) using numerical methods.
  • Checks for points where \( f'(x) \) is undefined (e.g., vertical tangents in \( \sqrt[3]{x} \)).
  • Supports an optional interval (e.g., \( [-2, 2] \)) to limit the search.
  • Keypad for easy input with digits, variable (\( x \)), operators (+, -, *, /, ^), and functions (sin, cos, ln, e^).
  • Clear and backspace functionality, with a "Copy" button for results.
• Practical Applications: Useful in calculus, optimization, and physics for analyzing function behavior.
• How to Use
  • Enter a function (e.g., \( x^3 - 3x + 1 \)) or a function with an interval (e.g., \( x^3 - 3x + 1; [-2, 2] \)).
  • Use the keypad to insert digits, variable (\( x \)), operators, and functions (sin, cos, ln, e^).
  • Click "Calculate" to find critical points and view steps, then use "Copy" to copy the result.
  • Use "Clear" to reset, or "⌫" to delete the last character.
  • Share or embed the calculator using the action buttons.
• Helpful Tips
  • Ensure the function is differentiable (e.g., avoid non-differentiable points like \( |x| \)).
  • Use * for multiplication and / for division in complex expressions.
  • Interval format: \( [a, b] \) with a semicolon (e.g., \( x^2; [-1, 1] \)).
• Examples
  • Example 1: Polynomial: \( x^3 - 3x + 1 \)
    • Input: \( x^3 - 3x + 1 \)
    • Steps: Compute \( f'(x) = 3x^2 - 3 \), solve \( 3x^2 - 3 = 0 \), roots at \( x = \pm 1 \).
    • Result: Critical points at \( x = -1, x = 1 \).
  • Example 2: Trigonometric: \( \sin(x); [0, 2\pi] \)
    • Input: \( \sin(x); [0, 2\pi] \)
    • Steps: Compute \( f'(x) = \cos(x) \), solve \( \cos(x) = 0 \), roots at \( x = \pi/2, 3\pi/2 \).
    • Result: Critical points at \( x = \pi/2, x = 3\pi/2 \).
  • Example 3: Rational Function: \( 1/x \)
    • Input: \( 1/x \)
    • Steps: Compute \( f'(x) = -1/x^2 \), no roots, undefined at \( x = 0 \).
    • Result: Critical point at \( x = 0 \) (undefined derivative).

Find critical points with detailed steps using this Critical Points Calculator. Share or embed it on your site!