Chapter 12: Q. 21 (page 976)

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

Short Answer

Expert verified

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

Step by step solution

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

Short Answer

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.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Calculate Discriminant.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Decision Maths

Probability and Statistics

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

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