**Problem Statement:**Find and test all critical points of $ f(x, y) $.**Function and Calculations:**The function is defined as:\[ f(x, y) = x^2 y + xy^2 - y \]The gradient of the function $ \nabla f $ is given by:\[ \nabla f = \langle 2xy + y^2, x^2 + 2xy - 1 \rangle \]The Hessian matrix $ H(f) $ is:\[ H(f) = \begin{bmatrix} 2y & 2x + 2y \\ 2x + 2y & 2x \end{bmatrix} \]**Explanation:**1. **Function:** The function $ f(x, y) $ is a polynomial involving both $ x $ and $ y $. 2. **Gradient ($ \nabla f $):** The gradient gives the vector of first partial derivatives of the function, which are essential for finding critical points where the gradient equals zero.3. **Hessian ($ H(f) $):** The Hessian matrix consists of the second partial derivatives, which help determine the nature of the critical points (i.e., whether they are minima, maxima, or saddle points).

Answered: Find and test all critical points of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Problem Statement:**

Find and test all critical points of $ f(x, y) $.

**Function and Calculations:**

The function is defined as:

\[ f(x, y) = x^2 y + xy^2 - y \]

The gradient of the function $ \nabla f $ is given by:

\[ \nabla f = \langle 2xy + y^2, x^2 + 2xy - 1 \rangle \]

The Hessian matrix $ H(f) $ is:

\[ H(f) = \begin{bmatrix} 2y & 2x + 2y \\ 2x + 2y & 2x \end{bmatrix} \]

**Explanation:**

1. **Function:** The function $ f(x, y) $ is a polynomial involving both $ x $ and $ y $.

2. **Gradient ($ \nabla f $):** The gradient gives the vector of first partial derivatives of the function, which are essential for finding critical points where the gradient equals zero.

3. **Hessian ($ H(f) $):** The Hessian matrix consists of the second partial derivatives, which help determine the nature of the critical points (i.e., whether they are minima, maxima, or saddle points).

Expert Solution

Step 1: Question Description

Find and test all critical points of $f open parentheses x comma space y close parentheses$ :

$f open parentheses x comma y close parentheses equals x squared y plus x y squared minus y$ , $nabla f equals less than 2 x y plus y squared comma space x squared plus 2 x y minus 1 greater than$ , $H open parentheses f close parentheses equals open square brackets table row cell 2 y end cell cell 2 x plus 2 y end cell row cell 2 x plus 2 y end cell cell 2 x end cell end table close square brackets$ .