Local Minimum

Definition

A local minimum is a value $f(c)$ that is less than or equal to all other function values in an open interval containing $c$.

• How to read: “The function value f of c is less than or equal to f of x for all x in a neighborhood of c.”
• Meaning: $f(c)$ is a valley relative to its immediate surroundings.

Why It Matters

Identifying local minima is critical for minimization problems, such as reducing cost, minimizing energy expenditure, or limiting error in a model. It helps in finding the most efficient or stable local state of a system.

Core Concepts

• Formal Condition: $f(c) \leq f(x)$ for all $x$ in some open interval $(c-\delta, c+\delta)$.
  • How to read: “f of c is less than or equal to f of x for all x near c.”
  • Meaning: $c$ is the lowest point in its local neighborhood.
• First Derivative Theorem: If $f$ has a local minimum at $c$ and $f'(c)$ exists, then $f'(c) = 0$.
  • How to read: “The derivative f prime of c equals zero.”
  • Meaning / when to use: The tangent line is horizontal at a local minimum (if the function is smooth).
• First Derivative Test: A local minimum occurs at a critical point $c$ if $f'(x)$ changes from negative to positive at $c$.
  • How to read: “The derivative changes sign from negative to positive.”
  • Meaning: The function was decreasing and starts increasing.

Connected Concepts

• Extrema
• Local Maximum
• Critical Points
• The First Derivative Test for Local Extrema
• Absolute Extreme Values