Q. 17 Give precise mathematical defini... [FREE SOLUTION]

Chapter 12: Q. 17 (page 987)

Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.
A function $f$ of two or three variables has a local minimum or a local maximum at a point $P$ .

Short Answer

Expert verified

(a) A function $f (x, y)$ has a local or relative maximum at $(x_{0}, y_{0})$ if $f (x_{0}, y_{0}) 鈮� f (x, y)$ for every point $(x, y)$ in some open disk containing $(x_{0}, y_{0})$ .

(b) A function $f (x, y)$ has a local or relative minimum at $(x_{0}, y_{0})$ if $f (x_{0}, y_{0}) 鈮� f (x, y)$ for every point $(x, y)$ in some open disk containing $(x_{0}, y_{0})$ .

Step by step solution

Step 1. Given information

A function $f$ of two or three variables has a local minimum or a local maximum at a point $P$ .

Step 2. Defining a local minimum or a local maximum at a point

(a) A function $f (x, y)$ has a local or relative maximum at $(x_{0}, y_{0})$ if $f (x_{0}, y_{0}) 鈮� f (x, y)$ for every point $(x, y)$ in some open disk containing $(x_{0}, y_{0})$ .

(b) A function $f (x, y)$ has a local or relative minimum at $(x_{0}, y_{0})$ if $f (x_{0}, y_{0}) 鈮� f (x, y)$ for every point $(x, y)$ in some open disk containing $(x_{0}, y_{0})$ .

Examples:

The function, $f (x, y) = \frac{1}{x^{2} + y^{2} 鈭� 1}$ has a local maximum at $(0, 0)$

The function, $f (x, y) = 8 x y + \frac{1}{x} + \frac{1}{y}$ has a local minimum at $(\frac{1}{2}, \frac{1}{2})$

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Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.
A function $f$ of two or three variables has a local minimum or a local maximum at a point $P$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Defining a local minimum or a local maximum at a point

One App. One Place for Learning.

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91影视

Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.A function fof two or three variables has a local minimum or a local maximum at a point P.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Defining a local minimum or a local maximum at a point

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Applied Mathematics

Logic and Functions

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.
A function $f$ of two or three variables has a local minimum or a local maximum at a point $P$ .