Question

Use the stationary properties of quadratic forms to determine the maximum and minimum values taken by the expression $$ Q=5 x^{2}+4 y^{2}+4 z^{2}+2 x z+2 x y $$ on the unit sphere \(x^{2}+y^{2}+z^{2}=1\). For what values of \(x, y, z\) do they occur?

Short answer

The maximum value is 5 at (±1, 0, 0). The minimum value is 4 at (0, ±1, 0) and (0, 0, ±1).

Step-by-step solution

- Define the problem with constraints

The goal is to maximize and minimize the quadratic form \[Q = 5x^2 + 4y^2 + 4z^2 + 2xz + 2xy\] subject to the constraint \[x^2 + y^2 + z^2 = 1.\] This requires using the method of Lagrange multipliers.

- Set up the Lagrangian function

Introduce a Lagrange multiplier \( \lambda \) and define the Lagrangian function: \[ \mathcal{L}(x, y, z, \lambda) = 5x^2 + 4y^2 + 4z^2 + 2xz + 2xy + \lambda (1 - x^2 - y^2 - z^2). \]

- Compute the partial derivatives

Find the partial derivatives of the Lagrangian function with respect to all variables and set them to zero: \[ \frac{\partial \mathcal{L}}{\partial x} = 10x + 2z + 2y - 2x\lambda = 0, \] \[ \frac{\partial \mathcal{L}}{\partial y} = 8y + 2x - 2y\lambda = 0, \] \[ \frac{\partial \mathcal{L}}{\partial z} = 8z + 2x - 2z\lambda = 0, \] \[ \frac{\partial \mathcal{L}}{\partial \lambda} = 1 - x^2 - y^2 - z^2 = 0. \] This gives us a system of equations to solve.

- Simplify and solve the system of equations

From the first three equations, solve for \( \lambda \): \[ \lambda = 5 + \frac{z}{x} + \frac{y}{x}, \] \[ \lambda = 4 + \frac{x}{y}, \] \[ \lambda = 4 + \frac{x}{z}. \] Equate the expressions for \( \lambda \) to find relationships between \( x, y, \) and \( z \).

- Identify critical points

By analyzing the above relationships, possible solutions must satisfy \[ x = \pm 1, \quad y = 0, \quad z = 0, \] \[ x = 0, \quad y = \pm 1, \quad z = 0, \] \[ x = 0, \quad y = 0, \quad z = \pm 1. \]

- Evaluate Q at each critical point

Calculate the value of \( Q \) at each critical point: For \( (1, 0, 0) \) and \( (-1, 0, 0) \): \[ Q = 5(1)^2 = 5. \] For \( (0, 1, 0) \) and \( (0, -1, 0) \): \[ Q = 4(1)^2 = 4. \] For \( (0, 0, 1) \) and \( (0, 0, -1) \): \[ Q = 4(1)^2 = 4. \]

- Determine maximum and minimum values

The maximum value of \( Q \) is 5, which occurs at \( (1, 0, 0) \) and \( (-1, 0, 0) \). The minimum value of \( Q \) is 4, which occurs at \( (0, 1, 0), (0, -1, 0), (0, 0, 1), \) and \( (0, 0, -1). \)

Key concepts

Stationary Properties

Stationary properties refer to points in a function where its rate of change is zero. For multivariable functions, these are the points where the gradient (derivative vector) is zero. To locate these stationary points, we use the method of Lagrange multipliers. This approach helps us find maximum and minimum values of a function subjected to constraints.
In the given exercise, the stationary properties of the quadratic form are explored to find these values subject to the constraint of lying on a unit sphere. Here, the constraint is expressed as \(x^2 + y^2 + z^2 = 1\). We introduce a Lagrange multiplier that helps us incorporate this constraint into a new function called the Lagrangian.
The Lagrangian function combines the original function and the constraint, turning our constrained optimization problem into an unconstrained one. We then compute partial derivatives concerning each variable and set them to zero. This gives us a system of equations whose solutions are the candidate points for the stationary properties. In simple words, one can think of Lagrange multipliers as a method of bending and stretching graph surfaces to meet the constraint while searching for peaks (maximum) and valleys (minimum).

Quadratic Forms

Quadratic forms are polynomial expressions where each term is of degree two. They appear in various fields like physics, engineering, and economics due to their ability to model phenomena involving squares of variables.
In this exercise, the quadratic form given is \Q = 5x^2 + 4y^2 + 4z^2 + 2xz + 2xy\. Here,

• The coefficients of each square term indicate their contribution to the form. For example, \(5x^2\) means the contribution of \(x^2\) is weighted by 5.
• The mixed terms like \2xz\ or \2xy\ explain how variables interact, affecting the overall value of the quadratic form.

Quadratic forms can represent geometrical shapes like ellipsoids or hyperboloids. The exercise aims to find the extreme values of this equation when applied on a unit sphere. This assessment requires optimizing the quadratic form under specific constraints, revealing how its terms interact to produce maximum and minimum values.
Recognizing the characteristics of quadratic forms helps in understanding their stationary properties, which is essential in determining extreme values. The quadratic form in the exercise represents a 3-dimensional shape, and our task is to pinpoint the peak and bottom values when confined to the unit sphere.

Unit Sphere Constraint

The unit sphere constraint restricts the variables \(x, y, z\) to lie on the surface of a sphere with radius 1, centered at the origin. Mathematically, it's expressed as \(x^2 + y^2 + z^2 = 1\). This means any solution must satisfy this constraint.
To solve the problem, we incorporate this constraint into our Lagrangian, changing the optimization problem from constrained to unconstrained. Our goal is to explore how the quadratic form behaves under this spherical restriction.
When finding the partial derivatives and setting them to zero, our solutions inherently satisfy the unit sphere constraint. The critical points determined (like \( (1, 0, 0)\) or \( (0, 1, 0)\)) are not just any points—they lie on the unit sphere. There, the quadratic form achieves its maximum and minimum constrained values.
This constraint is crucial because it narrows down potential solutions, ensuring that our extreme values fall on a more manageable set of points. Without it, solving for extreme values would be like searching in an open, infinite space, making the problem significantly more complicated.