How do you find the minimum and maximum values of a function?

To find the minimum and maximum values of a function, you typically follow these steps:

1. Identify the Function: Start with a function, say f(x), whose minimum and maximum values you want to determine.
2. Find the Derivative: Compute the derivative of the function, denoted as f'(x). This helps in finding critical points where the slope of the function is zero or undefined.
3. Set the Derivative to Zero: Solve the equation f'(x) = 0 to find the critical points. These points are potential candidates for local minima or maxima.
4. Evaluate Endpoints (if applicable): If the function is defined on a closed interval, make sure to also evaluate the function at the endpoints of that interval.
5. Determine the Nature of Critical Points: To classify the critical points, use the second derivative test or the first derivative test. The second derivative test involves computing f”(x): if f”(x) > 0 at a critical point, it is a local minimum; if f”(x) < 0, it is a local maximum.
6. Compare Values: Finally, compare the values of the function at all critical points and endpoints to determine the overall minimum and maximum values.

Keep in mind that these methods apply to differentiable functions. For functions that are not differentiable at certain points, you may need to consider those points carefully as potential local minima or maxima as well.