1. What is Global Maximum and Minimum?

The global maximum (or absolute maximum) of a function on an interval is the largest value that the function attains on that interval. Similarly, the global minimum is the smallest value. These are found by evaluating the function at critical points and endpoints.

2. How Does the Calculator Work?

The calculator finds global extrema by:

Example: For f(x) = x² on [-1, 2]:

• Critical point at x = 0
• Endpoint values at x = -1 and x = 2
• Global maximum is 4 at x = 2
• Global minimum is 0 at x = 0

3. Importance of Global Extrema

Applications: Global extrema are essential in optimization problems across physics, engineering, economics, and machine learning to find optimal solutions.

4. Using the Calculator

Tips: Enter a valid mathematical function and a closed interval [a,b]. The function should be continuous on the interval for reliable results.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between local and global extrema?
A: Local extrema are the highest/lowest points in a neighborhood, while global extrema are the absolute highest/lowest on the entire interval.

Q2: Can a function have multiple global maxima?
A: Yes, if the same maximum value occurs at multiple points (e.g., f(x) = sin(x) has infinitely many global maxima of 1).

Q3: What if my function isn't continuous?
A: The Extreme Value Theorem doesn't apply, and the function might not have global extrema on the interval.

Q4: How are critical points found?
A: By solving f'(x) = 0 or where f'(x) is undefined (but f(x) is defined).

Q5: What about multivariable functions?
A: This calculator handles single-variable functions only. Multivariable optimization requires different techniques.