MÁXIMOS, MÍNIMOS O PUNTOS DE SILLA. HESSIANO. Ejemplo 1. Video#135

Critical Points of a Function of Two Variables

Introduction to Critical Points

• The video introduces the concept of determining critical points for functions of two variables, aiming to classify them as local maxima, minima, or saddle points.

Problem Statement

• The function given for analysis is f(x,y) = x^2 + y^2 - 5x - 4y + xy . The task is to find its critical points and classify them.

Finding Partial Derivatives

• To find critical points, the process begins similarly to single-variable functions: derive the function and set it equal to zero.

• The first step involves calculating partial derivatives with respect to x and y .

Derivative with Respect to x

• For the partial derivative with respect to x , treat all other variables as constants. This results in:

• f_x = 2x - 5 + y .

Derivative with Respect to y

• Similarly, for the partial derivative with respect to y :

• f_y = 2y - 4 + x .

Setting Up Equations

• Next, both partial derivatives are set equal to zero:

• From f_x = 0: quad 2x - 5 + y = 0

• From f_y = 0: quad x + 2y - 4 = 0

Solving for Critical Points

• These equations form a system that can be solved simultaneously. Rearranging gives:

• First equation: y = 5 - 2x

• Second equation becomes: substituting into it leads us towards finding values for both variables.

Calculation Steps

• By manipulating these equations (multiplying one by two), we simplify and solve for values leading us toward identifying critical points.

Identifying Critical Point Values

• After calculations, we find a critical point at coordinates (2,1).

Classifying Critical Points Using Hessian Determinant

• To determine if this point is a maximum, minimum, or saddle point, we use the Hessian determinant defined as:

• D = f_xxf_yy - (f_xy)^2 .

Importance of Hessian Determinant

• Understanding how this determinant works will help classify our found critical point based on its value relative to zero.

Finding Second Derivatives and Critical Points

Steps to Calculate Second Derivatives

• The process begins with finding the second derivatives of a function, specifically focusing on f_xx , which involves taking the derivative of f_x again with respect to x .

• The first derivative yields 2x , while the derivatives of constants result in zero. Thus, we have established f_xx = 2 .

• Next, we calculate f_yy , which is the second derivative with respect to y . This results in a constant value of 2 for the term involving x , while other terms yield zero.

Substituting Values into Determinant Expression

• After determining both second derivatives, we substitute these values into a determinant expression. Here, it simplifies to calculating 4 - 1^2 = 3 .

• The determinant being greater than zero indicates that we are in a specific category concerning critical points.

Analyzing Critical Points

• With the determinant calculated as 3 (greater than zero), further analysis shows that if either condition holds true ( f_xx > 0), it confirms our findings.

• Conclusively, since both conditions are satisfied (determinant > 0 and f_xx > 0), this indicates that the critical point identified is indeed a minimum.

Conclusion