Determining Relative Minimums for a Function | Step-by-Step Guide

With respect to f(x), when do relative minimums occur?

A relative minimum occurs for a function f(x) at a specific point (x, f(x)) when the value of f(x) is lower than the values of nearby points on either side of that specific point

A relative minimum occurs for a function f(x) at a specific point (x, f(x)) when the value of f(x) is lower than the values of nearby points on either side of that specific point. In other words, a relative minimum is the lowest point in a small interval around that particular point.

To determine when a relative minimum occurs for a function, here are the steps:

1. Find the derivative of the function f(x) to obtain its derivative function f'(x).
2. Set f'(x) equal to zero and solve for x. These values of x are called critical points.
3. Determine the sign of f'(x) on either side of each critical point. You can do this by evaluating a test point in the interval on either side of the critical point into the derivative function f'(x).
– If f’ is positive to the left of the critical point and negative to the right, then it signifies a relative minimum.
– If f’ is negative to the left of the critical point and positive to the right, then it signifies a relative maximum.
– If f’ maintains the same sign on both sides of the critical point, it may represent an inflection point or a non-relative extremum.

4. After identifying the critical points that correspond to relative minimums, evaluate f(x) at these points to find the actual y-values of the relative minimums.

Note that if f(x) is defined on a closed interval [a, b], it is also crucial to consider the endpoints a and b while identifying relative minimums.

By following these steps, you can determine when relative minimums occur for a function f(x) and locate the points where they occur.

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