Finding the Absolute Minimum of a Function: Step-by-Step Guide and Tips

absolute min

the smallest y-value of the function, over the given interval (use x-values where first derivative = 0 & endpoints)

The absolute minimum of a function refers to the smallest value that the function can take on within a given interval or on its entire domain. In other words, it is the lowest point on the graph of the function.

To find the absolute minimum, you need to check all critical points and endpoints within the interval or domain. Critical points are points where the derivative of the function is either zero or undefined. Endpoints are the values at the boundaries of the interval or the limits of the domain.

Here is a step-by-step process to find the absolute minimum of a function:

1. Find all critical points by setting the derivative of the function equal to zero and solve for x. Also, consider any points where the derivative is undefined (e.g., points where the function has vertical asymptotes).

2. Calculate the function values of the critical points obtained in step 1.

3. Determine the function values at the endpoints of the interval or domain. Plug in the values of the endpoints into the function.

4. Compare the function values obtained in steps 2 and 3. The smallest value represents the absolute minimum.

It is important to note that the absolute minimum might occur at one or more points within the interval or domain. If the function is continuous, the absolute minimum will always exist.

Keep in mind that the absolute minimum refers to the function’s values, not its coordinates. To find the coordinates (x, y) of the absolute minimum, you need to determine the corresponding x-values obtained in step 4 and then substitute them back into the original function to calculate the y-values.

More Answers: