Find all points (x, y) where f(x, y) has a possible relative...

Find all points $(x, y)$ where $f(x, y)$ has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of $f(x, y)$ at each of these points. If the second-derivative test is inconclusive, so state. $$f(x, y)=x^{2}-2 x y+3 y^{2}+4 x-16 y+22$$

Key Concepts

Critical Points

Critical points are the locations in the domain of a function where the gradient (or vector of first partial derivatives) is zero or undefined. These points are candidates for local extreme values (relative minima or maxima) or saddle points, and identifying them is a crucial step in optimization problems for functions of several variables.

Gradient

The gradient is a vector that contains the first partial derivatives of a function with respect to its variables. It points in the direction of the steepest ascent of the function, and setting this vector equal to zero is used to find the function’s critical points, which are crucial for determining local maxima, minima, or saddle points.

Hessian Matrix

The Hessian matrix is a square matrix composed of the second partial derivatives of a function. This matrix encapsulates the local curvature of the function and is used in the second-derivative test, which helps determine whether a critical point is a local minimum, local maximum, or a saddle point for functions of multiple variables.

Second-Derivative Test

The second-derivative test is a method used to classify critical points in multivariable calculus. By evaluating the determinant and, in some cases, the signs of the eigenvalues of the Hessian matrix at a critical point, one can conclude if the point corresponds to a local minimum, a local maximum, or a saddle point. However, if the test is inconclusive (for example, when the determinant is zero), additional investigation is required to determine the nature of the critical point.

Quadratic Forms

Quadratic forms are expressions involving squared variables, often written in matrix notation, and they appear in the study of second-order approximations of functions near critical points. Analysis of the quadratic form, for instance by completing the square or by studying the associated Hessian matrix, provides insight into the local behavior of the function, helping to classify the type of critical point.

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