Question

What is a stationary point of a function of two variables, $f(x, y)$ ? What, if anything, is the difference between a critical point and a stationary point of $f ?$

Name: Problem 4, Chapter 12: Calculus
Uploaded: 2022-02-28T20:11:37-08:00
Duration: 3 min 48 s
Channel: Muhammad Saleem
Description: What is a stationary point of a function of two variables, f(x, y) ? What, if anything, is the difference between a critical point and a stationary point of f ?

   What is a stationary point of a function of two variables, $f(x, y)$ ? What, if anything, is the difference between a critical point and a stationary point of $f ?$

Calculus

Laura Taalman, Peter… 1st Edition

Chapter 12, Problem 4

Instant Answer

Solved by Expert Muhammad Saleem

02/28/2022

Step 1

A stationary point of a function of two variables, $f(x, y)$, is a point in the domain of $f(x, y)$ at which the function is differentiable and the gradient of the function vanishes. Mathematically, this is represented as $\nabla f(x, y) = 0$. Show more…

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What is a stationary point of a function of two variables, $f(x, y)$ ? What, if anything, is the difference between a critical point and a stationary point of $f ?$

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Key Concepts

Stationary Point

A stationary point of a function of two variables is a point in its domain where all the first-order partial derivatives exist and are equal to zero. In other words, it is a point at which the gradient vector of the function vanishes. This concept is important because at these points the function does not increase or decrease in any infinitesimal direction, making them potential candidates for local minima, local maxima, or saddle points.

Critical Point

A critical point of a function is more broadly defined as any point in the domain where the function’s derivative information is 'degenerate.' This includes points where the gradient vanishes (stationary points) as well as points where one or more partial derivatives do not exist. Thus, while every stationary point is a critical point, not every critical point is a stationary point, since critical points may also occur where the function is non-differentiable.

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