Local Minimum

Definition

A local minimum is a point on a function's graph where the function value is less than or equal to the function values in the immediate surrounding area. It represents a point where the function attains a minimum value within a localized region, even if it is not the absolute minimum value of the function over its entire domain.

5 Must Know Facts For Your Next Test

1. Local minima are important in optimization problems, where the goal is to find the point that minimizes or maximizes a function.
2. The first derivative test can be used to determine if a critical point is a local minimum, local maximum, or neither.
3. Local minima can exist on both continuous and discrete functions, and their identification is crucial in many real-world applications.
4. The Second Derivative Test can also be used to classify critical points as local minima, local maxima, or saddle points.
5. In multivariable functions, local minima are found at points where all the partial derivatives are zero, and the Hessian matrix is positive definite.

Review Questions

• Explain the difference between a local minimum and a global minimum, and describe how they are identified on a function's graph.
  • A local minimum is a point on a function's graph where the function value is less than or equal to the function values in the immediate surrounding area, but it may not be the absolute lowest value of the function over its entire domain. In contrast, a global minimum is the point on the function's graph where the function attains its absolute lowest value over the entire domain. Local minima can be identified by looking for points where the function changes from decreasing to increasing, while global minima are the lowest point on the entire function graph.
• Describe how the first and second derivative tests can be used to classify critical points as local minima, local maxima, or neither.
  • The first derivative test can be used to determine if a critical point is a local minimum, local maximum, or neither. If the first derivative changes from positive to negative at the critical point, then the critical point is a local maximum. If the first derivative changes from negative to positive at the critical point, then the critical point is a local minimum. If the first derivative does not change sign at the critical point, then the critical point is neither a local maximum nor a local minimum. The second derivative test can also be used to classify critical points, where a critical point is a local minimum if the second derivative is positive, a local maximum if the second derivative is negative, and neither a local minimum nor maximum if the second derivative is zero.
• Explain the importance of local minima in optimization problems and describe how they are identified for multivariable functions.
  • Local minima are crucial in optimization problems, where the goal is to find the point that minimizes or maximizes a function. Identifying local minima is important because they represent points where the function attains a minimum value within a localized region, even if it is not the absolute minimum value of the function over its entire domain. For multivariable functions, local minima are found at points where all the partial derivatives are zero, and the Hessian matrix, which contains the second partial derivatives, is positive definite. This ensures that the critical point is a local minimum and not a local maximum or saddle point. The identification of local minima is essential in many real-world applications, such as engineering design, resource allocation, and financial portfolio optimization.

Related terms

Global Minimum: The point on a function's graph where the function attains its absolute lowest value over the entire domain of the function.

Critical Point: A point on a function's graph where the derivative of the function is equal to zero or undefined, indicating a possible location for a local maximum or minimum.

Convex Function: A function where the line segment between any two points on the graph lies on or above the graph, resulting in a function with no local minima other than the global minimum.