1. What is a Critical Point?

A critical point of a function is a point where the derivative is either zero or undefined. These points are important in determining the behavior of functions, particularly for finding local maxima and minima.

2. How to Find Critical Points

The process to find critical points:

Example: For f(x) = x² - 4x + 4:

• f'(x) = 2x - 4
• Set 2x - 4 = 0 → x = 2
• Critical point at x = 2

3. Importance of Critical Points

Details: Critical points help identify potential local maxima, minima, or inflection points. They are essential in optimization problems across mathematics, physics, and engineering.

4. Using the Calculator

Tips: Enter your function in terms of x (e.g., "x^3 - 3x^2"). The calculator will find where the derivative equals zero or is undefined.

5. Frequently Asked Questions (FAQ)

Q1: Are all critical points maxima or minima?
A: No, some critical points are inflection points or saddle points where the function doesn't have a maximum or minimum.

Q2: How do I determine if a critical point is max, min, or neither?
A: Use the second derivative test or analyze the sign change of the first derivative around the critical point.

Q3: Can a function have multiple critical points?
A: Yes, functions can have zero, one, or many critical points depending on their complexity.

Q4: What if the derivative is undefined at a point?
A: Points where the derivative is undefined (like sharp corners) are also considered critical points.

Q5: Are critical points the same as zeros of a function?
A: No, critical points are where the derivative is zero/undefined, while zeros are where the function itself equals zero.