Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

In Exercises 21–26, find the discriminant of the given function.
$f (x, y) = e^{2 x} \cos y$ .

The answer is $- 4 e^{4 x}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information.

The function is $f (x, y) = e^{2 x} \cos y$ .

Step 2. Explanation.

The discriminant is calculated by formula,

$d e t (H_{f}) = \frac{\partial^{2} f}{\partial x^{2}} \cdot \frac{\partial^{2} f}{\partial y^{2}} - {(\frac{\partial^{2} f}{\partial x \partial y})}^{2}$

Find $\frac{\partial f}{\partial x}$ , $\frac{\partial^{2} f}{\partial x^{2}}$ , $\frac{\partial f}{\partial y}$ , $\frac{\partial^{2} f}{\partial y^{2}}$ and localid="1649783903210" $\frac{\partial^{2} f}{\partial x \partial y}$

$\frac{\partial f}{\partial x} = 2 e^{2 x} \cos y$ , $\frac{\partial^{2} f}{\partial x^{2}} = 4 e^{2 x} \cos y$

$\frac{\partial f}{\partial y} = - e^{2 x} \sin y$ , localid="1649783850374" $\frac{\partial^{2} f}{\partial y^{2}} = - e^{2 x} \cos y$

localid="1649818601411" $\frac{\partial^{2} f}{\partial x \partial y} = - 2 e^{2 x} \sin y$

Step 3. Calculate Discriminant.

Calculate $d e t (H_{f}) = \frac{\partial^{2} f}{\partial x^{2}} \cdot \frac{\partial^{2} f}{\partial y^{2}} - {(\frac{\partial^{2} f}{\partial x \partial y})}^{2}$

localid="1649818688351" $d e t (H_{f}) = (4 e^{2 x} \cos y) (- e^{2 x} \cos y) - {(- 2 e^{2 x} \sin y)}^{2} = - 4 e^{4 x} \cos^{2} y - 4 e^{4 x} \sin^{2} x = - 4 e^{4 x}$

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Calculus

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Short Answer

In Exercises 21–26, find the discriminant of the given function.
$f (x, y) = e^{2 x} \cos y$ .

Step by Step Solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Calculate Discriminant.

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In Exercises 21–26, find the discriminant of the given function.fx,y=e2xcosy.

Step by Step Solution

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Step 2. Explanation.

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In Exercises 21–26, find the discriminant of the given function.
$f (x, y) = e^{2 x} \cos y$ .