A local minimum of a function is a point where the function value is lower than at any nearby points. Mathematically, if $f(c)$ is a local minimum, then $f(c) \leq f(x)$ for all $x$ in some interval around $c$.
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A local minimum occurs at a critical point where the first derivative of the function is zero or undefined.
The second derivative test can be used to determine if a critical point is a local minimum; if $f''(c) > 0$, then $f(c)$ is a local minimum.
Not all critical points are local minima; they could also be local maxima or saddle points.
A graph of the function near a local minimum will show the curve dipping down to the minimum point and then rising back up.
Local minima are important in optimization problems where you want to find the lowest value of a function within a certain range.
Review Questions
What conditions must be met for a point to be considered a local minimum?
How does the second derivative test help identify a local minimum?
Can you have more than one local minimum in a single function? Explain why or why not.
Related terms
Critical Point: A point on the graph of a function where its first derivative is zero or undefined.
Second Derivative Test: A method used to determine whether a critical point is a local maximum, minimum, or saddle point based on the sign of the second derivative.
Local Maximum: A point where the function value is higher than at any nearby points.