Q. 19 Give precise mathematical defini... [FREE SOLUTION]

Chapter 12: Q. 19 (page 988)

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 stationary point at a point $P$ .

Short Answer

Expert verified

A function $f$ of two or three variables has a stationary point at a point $P$ in the domain of the function $f$ if $f$ is differentiable at $P$ and gradient of $f$ at $P$ is zero.

Step by step solution

Step 1. Given information

A function $f$ of two or three variables has a stationary point at a point $P$ .

Step 2. Defining stationary point

A function $f$ of two or three variables has a stationary point at a point $P$ in the domain of a function $f$ if $f$ is differentiable at $P$ and gradient of $f$ at $P$ is zero.

Example:

For a function, $f (x, y) = 3 x^{2} + 3 x + 6 y^{2} 鈭� 7$

The first order partial derivatives are: $f_{x} (x, y) = \frac{鈭� f}{鈭� x} = 6 x + 3 and f_{y} (x, y) = \frac{鈭� f}{鈭� y} = 12 y$

Solving the above system of equations, we, get, $x = - \frac{1}{2}$ and $y = 0$

The point, $(- \frac{1}{2}, 0)$ is a stationary point.

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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 stationary point at a point $P$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Defining stationary point

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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 stationary point at a point P.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Defining stationary point

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Pure Maths

Geometry

Mechanics Maths

Applied Mathematics

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 stationary point at a point $P$ .