Question

Extreme points on flattened spheres The equation \(x^{2 n}+y^{2 n}+z^{2 n}=1,\) where \(n\) is a positive integer, describes a flattened sphere. Define the extreme points to be the points on the flattened sphere with a maximum distance from the origin. a. Find all the extreme points on the flattened sphere with \(n=2 .\) What is the distance between the extreme points and the origin? b. Find all the extreme points on the flattened sphere for integers \(n>2 .\) What is the distance between the extreme points and the origin? c. Give the location of the extreme points in the limit as \(n \rightarrow \infty\) What is the limiting distance between the extreme points and the origin as \(n \rightarrow \infty ?\)

Short answer

To summarize, for a flattened sphere with \(x^{2n} + y^{2n} + z^{2n} = 1\), the extreme points are always located on the xy-plane, having a distance of 1 from the origin. When n=2, the extreme points lie along a circle. For integers n>2 and as n approaches infinity, the extreme points become concentrated along the x and y axes, approaching the union of the lines x=1, y=1, x=-1, and y=-1. The limiting extreme points are (1,0,0), (0,1,0), (-1,0,0), and (0,-1,0), which all have a distance of 1 from the origin.

Step-by-step solution

Plug n=2 into the equation and find extreme points

Let's first plug in n=2 into the equation: \(x^{2\cdot2} + y^{2\cdot2} + z^{2\cdot2} = 1 \Rightarrow x^4 + y^4 + z^4 = 1\). To find the extreme points, we can set one of the variables equal to zero and then solve for the other variables.

Solve for x, y, and z when one variable is zero

Let's calculate x, y, and z when one of the variables is equal to zero: a) x=0: in this case, \(y^4 + z^4 = 1 \Rightarrow (y^2 + z^2)(y^2 - z^2) = 1\). Since y and z are real numbers, their squares are always non-negative, so the product is always non-negative. Therefore, there can't be any extreme points with x=0. b) y=0: the same argument as before applies. There can't be any extreme points with y=0. c) z=0: in this case, \(x^4 + y^4 = 1 \Rightarrow x^2 + y^2 = 1\), which is a circle in the xy-plane.

Find the points on the circle with the maximum distance from the origin

From the circle equation, we can see that any point on the circle has the distance of 1 from the origin (since \(x^2 + y^2 = 1\)). Therefore, all points on the circle with z=0 are extreme points, and their distance from the origin is 1. #b. Find all the extreme points on the flattened sphere for integers n>2. What is the distance between the extreme points and the origin?#

Extend the reasoning from the case n=2

By following similar steps as in the case n=2, we can see that there will be no extreme points when x=0 or y=0. When z=0, we'll get the equation \(x^{2n} + y^{2n} = 1\).

Explain why this equation represents all the extreme points

Since the powers of x and y are even and z=0, all the points satisfying this equation will be on the xy-plane and have a distance of 1 from the origin. Therefore, all the extreme points on the flattened sphere for integers n>2 are represented by the equation \(x^{2n} + y^{2n} = 1\) for z=0, and their distance from the origin is 1. #c. Give the location of the extreme points in the limit as n approaches infinity. What is the limiting distance between the extreme points and the origin as n approaches infinity?#

Discuss the limit of the equation as n approaches infinity

As n approaches infinity, the equation \(x^{2n} + y^{2n} = 1\) for z=0 becomes more and more concentrated along the x and y axes, approaching the union of the lines x=1, y=1, x=-1, and y=-1.

Give the limiting location of the extreme points

Therefore, the location of the extreme points in the limit as n approaches infinity are the points where the lines x=1, y=1, x=-1, and y=-1 intersect the xy-plane, which are (1,0,0), (0,1,0), (-1,0,0), and (0,-1,0).

Calculate the limiting distance between the extreme points and the origin

The limiting distance between the extreme points and the origin as n approaches infinity is 1 (since they are on the lines x=1, y=1, x=-1, and y=-1).

Key concepts

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus to functions with more than one variable. It involves multiple dimensions and can be used to study the relationships between these variables in space. For instance, in the problem of finding extreme points on a flattened sphere given by the equation x^{2n} + y^{2n} + z^{2n} = 1, multivariable calculus allows us to investigate points in three-dimensional space. To find the extreme points or the maximum distance from the origin (0,0,0), we utilize partial derivatives, gradients, and critical points, which are fundamental tools in multivariable calculus.

By examining different values of n and discussing cases where one or more variables are zero, we actively employ multivariable calculus concepts to solve the given problem. Concepts such as continuity, differentiation of functions with several variables, and optimization in higher dimensions all play a crucial role in approaching and solving such problems.

Distance from Origin

The distance from the origin in a Cartesian coordinate system is a fundamental concept when examining points in space. Mathematically, this distance can be calculated using the Pythagorean theorem extended to three dimensions. For a point (x, y, z), the distance from the origin is given by d = \(\sqrt{x^2 + y^2 + z^2}\).

In our exercise, determining the extreme points of the flattened sphere involves finding points that maximize this distance. However, it's interesting to note that for the flattened sphere equations given, especially when z=0, this distance remains constant at 1. This constant value arises from the equation being normalized, meaning all points satisfying the equation are equidistant from the origin in the context of the sphere's surface, a crucial property influencing the symmetry and shape of the flattened sphere.

Limit of a Function

The limit of a function is a foundational concept in calculus used to describe the behavior of a function as its input approaches a particular value. Limits can be finite or infinite and are essential for defining continuity, derivatives, and integrals. In our specific case, analyzing the limit as n approaches infinity for the equation of the flattened sphere reveals the function's end-behavior.

For large values of n, each term in the equation x^{2n} + y^{2n} + z^{2n} = 1 tends to become more concentrated around the axes, which eventually leads to the function resembling a cross-like shape along the x and y axes when z=0. This approximation aids in understanding how the shape of the flattened sphere evolves and ultimately determines the location of its extreme points in the limit — the points (1,0,0), (0,1,0), (-1,0,0), and (0,-1,0) with a distance of 1 from the origin.

Symmetry in Equations

Symmetry in equations often implies that a system has a balanced or harmonious arrangement that can be exploited to simplify analysis or computation. In the context of our exercise, the symmetry of the equation x^{2n} + y^{2n} + z^{2n} = 1 indicates invariance under certain transformations. Specifically, this equation is symmetric with respect to rotations around the origin and reflections through planes that intersect the origin.

Exploring symmetry allows us to make critical observations about the potential location of extreme points. For example, by recognizing that the equation is unchanged when x, y, or z is replaced with their negatives, we infer that extreme points occur in symmetric pairs or groups. The existence of such symmetries particularly simplifies finding solutions, as some can be deduced directly from others. Overall, symmetries underscore the geometric nature of the solutions and lead to a deeper understanding of the spatial distribution of extreme points on these unique surfaces.