Critical Points of Multivariable Functions

Definition

A critical point $(a, b)$ is an interior point in the domain of $f(x, y)$ where either the gradient is zero ($\nabla f = \mathbf{0}$) or the gradient does not exist.

Why It Matters

Multivariable critical points extend optimization to complex, multi-dimensional systems. They are essential for solving real-world problems where multiple factors interact, such as maximizing profit in a business or minimizing energy in a physical system.

Core Concepts

• Necessary Condition: If $f$ has a local extreme at $(a, b)$, then $(a, b)$ must be a critical point.
• Zero Gradient: $f_x(a, b) = 0$ AND $f_y(a, b) = 0$. This implies the tangent plane is horizontal.
  • How to read: “The partial derivative f x at a, b equals zero, and the partial derivative f y at a, b equals zero.”
  • Meaning / when to use: Solve this system to find candidate extrema and saddle points; tangent plane is horizontal.
• Candidates: Critical points are the “suspects” for local maxima, minima, or saddle points.

Connected Concepts

• Critical Points
• Chain Rule for Multivariable Functions
• Distance Between Points in Space