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Find All Critical Points of The Following Function Calculator

Reviewed by Calculator Editorial Team

Finding critical points of a function is essential in calculus for identifying maxima, minima, and saddle points. This calculator helps you find all critical points of a given function by solving for where the derivative equals zero or is undefined.

What Are Critical Points?

Critical points are values of x where the derivative of a function is either zero or undefined. These points are potential locations for local maxima, local minima, or saddle points.

For a function f(x), a critical point occurs at x = c if either:

  • f'(c) = 0 (the derivative is zero)
  • f'(c) is undefined (the derivative does not exist)

Note: Critical points are not necessarily maxima or minima. Further analysis is required to classify them.

How to Find Critical Points

To find critical points of a function:

  1. Find the first derivative of the function f(x).
  2. Set the derivative equal to zero: f'(x) = 0.
  3. Solve for x to find potential critical points.
  4. Check for points where the derivative is undefined.

Example: Find critical points of f(x) = x³ - 3x² + 4.

1. First derivative: f'(x) = 3x² - 6x.

2. Set derivative to zero: 3x² - 6x = 0.

3. Solve: x(3x - 6) = 0 → x = 0 or x = 2.

Critical points are at x = 0 and x = 2.

How to Use This Calculator

Enter your function in the input field below. The calculator will:

  1. Compute the first derivative of your function.
  2. Find all values of x where the derivative is zero or undefined.
  3. Display the critical points and the derivative function.
  4. Show a graph of the function and its derivative for visualization.

Supported functions include polynomials, trigonometric functions, exponentials, and logarithms.

Interpreting Results

After finding critical points, you should:

  1. Classify each critical point as a maximum, minimum, or saddle point using the second derivative test or first derivative test.
  2. Consider the behavior of the function around each critical point.
  3. Check for any restrictions on the domain of the function.
Test Condition Conclusion
First Derivative Test Sign change of f'(x) around c Local max if changes + to -, min if - to +
Second Derivative Test f''(c) > 0 Local minimum
Second Derivative Test f''(c) < 0 Local maximum

FAQ

What is the difference between critical points and inflection points?
Critical points are where the derivative is zero or undefined. Inflection points are where the concavity changes, typically found by analyzing the second derivative.
Can this calculator handle functions with multiple variables?
No, this calculator is designed for single-variable functions. For multivariable functions, you would need partial derivatives.
What if the derivative is undefined at a point but not zero?
That point is still a critical point. For example, f(x) = |x| has a critical point at x = 0 where the derivative does not exist.