What is the minimum and maximum values of the following...

What is the minimum and maximum values of the following rational functions:
(a) $\frac{3 x^2-2 y^2}{x^2+y^2}$,
(b) $\frac{x^2-3 x y+y^2}{x^2+y^2}$,
(c) $\frac{3 x^2+x y+5 y^2}{x^2+y^2}$,
(d) $\frac{2 x^2+x y+3 x z+2 y^2+2 z^2}{x^2+y^2+z^2}$

Key Concepts

Constrained Optimization Using Lagrange Multipliers

Constrained optimization techniques, such as the method of Lagrange multipliers, are used to find the extreme values of functions subject to a constraint (for example, fixing the norm of the vector variables). Since the expressions in question are invariant under scaling of the variables, one can impose a normalization condition (like x²+y²=1), and then apply Lagrange multipliers to convert the problem into finding stationary points of a Lagrangian, which is a standard method in optimization theory.

Polar Coordinates and Trigonometric Substitution

For functions defined on two variables with a rotationally symmetric denominator like x²+y², converting to polar coordinates (substituting x = r cos? and y = r sin?) is a highly useful technique. This substitution leverages the symmetry of the denominator and reduces the problem to one variable (?), often simplifying the process of determining the maximum and minimum values of the function.

Rayleigh Quotient

The Rayleigh quotient is a concept that arises when studying the ratio of two quadratic forms. It expresses a quantity of the form (x?Ax)/(x?x), where A is a symmetric matrix. The extreme values of the Rayleigh quotient correspond to the smallest and largest eigenvalues of the matrix A. This approach is crucial when optimizing ratios that are scale invariant, as it transforms the problem of finding maximum and minimum values into an eigenvalue problem.

Quadratic Forms

Quadratic forms are polynomial expressions in which variables appear only to the second degree, often in the form ax² + bxy + cy² (or their extensions in higher dimensions). In these problems, both the numerator and denominator are quadratic forms. Analyzing such expressions is central to understanding how the function behaves, and it connects directly with techniques such as matrix representation and diagonalization.

Homogeneous Functions

Homogeneous functions are functions in which every term is of the same degree. In the context of these rational functions, the numerator and the denominator are both quadratic forms (i.e., homogeneous of degree 2), which means the overall function is homogeneous of degree 0. This property allows one to simplify the problem by setting a condition such as x²+y² = 1 (or its equivalent), reducing the number of independent variables and making it easier to analyze the extrema.

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