Local minimum

Definition

A local minimum of a function is a point where the function value is lower than at all nearby points. It represents the lowest value within a specific interval or neighborhood.

5 Must Know Facts For Your Next Test

1. A local minimum occurs where the first derivative of the function changes from negative to positive.
2. The second derivative test can confirm a local minimum if it is greater than zero at that point.
3. A polynomial function can have multiple local minima depending on its degree.
4. Local minima are critical points but not all critical points are local minima.
5. They are useful in optimization problems to identify the least cost, energy, or other quantities.

Review Questions

• What condition must the first derivative satisfy at a local minimum?
• How does the second derivative test confirm a local minimum?
• Can a polynomial function have more than one local minimum? Explain why.

Related terms

Critical Point: A point on the graph of a function where its derivative is zero or undefined.

Global Minimum: The smallest value of a function over its entire domain.

Second Derivative Test: A method used to determine if a critical point is a local maximum, minimum, or saddle point by evaluating the sign of the second derivative at that point.