Local minimum

Definition

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

5 Must Know Facts For Your Next Test

1. A local minimum occurs where the first derivative, $f'(x)$, changes from negative to positive.
2. At a local minimum, the first derivative equals zero or does not exist.
3. The second derivative test can confirm if a critical point is a local minimum: if $f''(x) > 0$, it's a local minimum.
4. Local minima are important for understanding the behavior of polynomial functions within certain intervals.
5. Graphically, a local minimum appears as the bottom point of a valley on the graph of the function.

Review Questions

• What condition must be met by the first derivative at a local minimum?
• How can you use the second derivative to confirm that a critical point is a local minimum?
• Explain how you would identify a local minimum on the graph of a polynomial function.

Related terms

Critical Point: A point on the graph where the first derivative is zero or undefined and potential locations for local maxima and minima.

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

Inflection Point: A point on the curve where the concavity changes, indicated by changes in sign in $f''(x)$.