Critical Points

Definition

A critical point is an interior point in the domain of a function where the derivative is either zero ($f'(c) = 0$) or undefined. These points are the primary “candidates” for local extrema.

Why It Matters

Critical points are the primary tools for optimization in mathematics and science. They allow us to find the peaks and valleys of any system, identifying the exact moments of maximum efficiency or minimum cost.

Core Concepts

• Horizontal Tangents: Points where the rate of change is zero, indicating a potential peak or valley.
  • How to read: “The derivative f prime of c equals zero.”
  • Meaning: Slope vanishes—possible local max, min, or inflection (not guaranteed).
• Singularities: Points where the derivative fails to exist (corners, cusps, or vertical tangents).
• Candidate Test: To find absolute extrema on $[a, b]$, one must test all critical points and both endpoints ($a$ and $b$).

Connected Concepts

• Critical Points of Multivariable Functions
• Distance Between Points in Space
• Points of Inflection