Local minimum

Definition

A local minimum refers to a point in a function where the value of the function is lower than the values of the function at nearby points. This concept is crucial in optimization as it helps identify the lowest points within a specific neighborhood of the function, rather than the lowest point overall, known as a global minimum. Identifying local minima is essential for understanding how functions behave and for finding optimal solutions in various applications.

5 Must Know Facts For Your Next Test

1. A local minimum can be identified using the first derivative test, where if the derivative changes from negative to positive at that point, it indicates a local minimum.
2. Local minima may not be unique; there can be multiple local minima within the same function.
3. In multivariable functions, a local minimum may require evaluating partial derivatives to identify critical points.
4. Not all critical points are local minima; some could be local maxima or saddle points, depending on the behavior of the function.
5. The shape of the function's graph around a local minimum will typically appear like a 'valley' in the region surrounding that point.

Review Questions

• How can you identify a local minimum using calculus techniques?
  • To identify a local minimum, you can use the first derivative test. First, find the critical points by setting the first derivative of the function equal to zero. Then, check the sign of the derivative before and after each critical point. If the derivative changes from negative to positive, that critical point is a local minimum.
• What role does the second derivative test play in determining whether a critical point is a local minimum?
  • The second derivative test helps determine if a critical point is a local minimum by evaluating the second derivative at that point. If the second derivative is positive at that critical point, it indicates that the function is concave up, confirming that it is indeed a local minimum. If it is negative, that point would be a local maximum.
• Compare and contrast local minima and global minima in terms of their importance in optimization problems.
  • Local minima and global minima serve different purposes in optimization problems. A global minimum represents the absolute lowest point across an entire function, which is ideal when searching for optimal solutions. In contrast, local minima are important for identifying optimal solutions within specific intervals or constraints. Sometimes, optimization may only need to find a good enough solution quickly, which can lead to identifying a local minimum instead of exhaustively searching for the global minimum. Understanding both concepts is key for effective problem-solving in economics and other fields.

Related terms

Global minimum: The lowest point of a function over its entire domain, as opposed to just a local neighborhood.

Critical point: A point in the domain of a function where its derivative is either zero or undefined, often used to find local minima and maxima.

Second derivative test: A method used to determine whether a critical point is a local minimum, local maximum, or neither by evaluating the second derivative at that point.