Finding critical points is key to understanding a function's behavior. We use partial derivatives to locate these points where the function might reach its highest or lowest values, or change direction.

The second derivative test helps classify these critical points. By examining the Hessian matrix, we can determine if a point is a local maximum, minimum, or saddle point, revealing the function's shape in that area.

Critical Points and Partial Derivatives

Identifying Critical Points

• Critical points occur where the partial derivatives of a multivariable function are simultaneously zero or undefined
• To find critical points, set each partial derivative equal to zero and solve the resulting system of equations
• Critical points represent potential local maxima, local minima, or saddle points of the function

Gradient Vector and its Applications

• The gradient vector is a vector-valued function that points in the direction of steepest ascent of a scalar-valued function
• Partial derivatives are the components of the gradient vector, representing the rates of change of the function with respect to each variable
• The gradient vector is perpendicular to the level curves or level surfaces of the function at any given point
• The magnitude of the gradient vector indicates the steepness of the function at a point (larger magnitude implies steeper slope)

Classifying Critical Points

Hessian Matrix and its Determinant

• The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function
• For a function $f(x, y)$, the Hessian matrix is given by: $H(x, y) = \begin{bmatrix} f_{xx}(x, y) & f_{xy}(x, y) \\ f_{yx}(x, y) & f_{yy}(x, y) \end{bmatrix}$
• The determinant of the Hessian matrix, denoted as $\det(H)$, helps classify critical points
• If $\det(H) > 0$, the critical point is either a local maximum or a local minimum
• If $\det(H) < 0$, the critical point is a saddle point

Second Derivative Test for Classification

• The second derivative test uses the Hessian matrix to classify critical points
• For a critical point $(a, b)$:
  • If $\det(H(a, b)) > 0$ and $f_{xx}(a, b) < 0$, the point is a local maximum
  • If $\det(H(a, b)) > 0$ and $f_{xx}(a, b) > 0$, the point is a local minimum
  • If $\det(H(a, b)) < 0$, the point is a saddle point
• A local maximum occurs when the function decreases in all directions from the critical point (peaks or hills)
• A local minimum occurs when the function increases in all directions from the critical point (valleys or basins)
• A saddle point occurs when the function increases in some directions and decreases in others from the critical point (resembles a saddle or mountain pass)