Chapter 12: Q. 22 (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.
The Hessian and discriminant of a function $f (x, y)$ .

Short Answer

Expert verified

Let $f (x, y)$ be a function with continuous second-order partial derivatives on some open set $S$ .

(a) The Hessian of $f$ is the 2 × 2 matrix of second-order partial derivatives: $H_{f} = [\begin{matrix} \frac{\partial^{2} f}{\partial x^{2}} & \frac{\partial^{2} f}{\partial y \partial x} \\ \frac{\partial^{2} f}{\partial x \partial y} & \frac{\partial^{2} f}{\partial y^{2}} \end{matrix}]$

(b) The discriminant of $f$ is the determinant of the Hessian. That is, $d e t (H_{f}) = \frac{\partial^{2} f}{\partial x^{2}} \cdot \frac{\partial^{2} f}{\partial y^{2}} - {(\frac{\partial^{2} f}{\partial x \partial y})}^{2}$

Step by step solution

Step 1. Given information

The Hessian and discriminant of a function $f (x, y)$ .

Step 2. Defining hessian and discriminant

Let $f (x, y)$ be a function with continuous second-order partial derivatives on some open set $S$ .

Example:

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

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

The second-order partial derivatives are: $f_{x x} (x, y) = \frac{\partial^{2} f}{\partial x^{2}} = 6, f_{y y} (x, y) = \frac{\partial^{2} f}{\partial y^{2}} = 12 and f_{x y} (x, y) = \frac{\partial^{2} f}{\partial x \partial y} = 0$

The Hessian is: $H_{f} = [\begin{matrix} 6 & 0 \\ 0 & 12 \end{matrix}]$

The discriminant of the function or determinant of the hessian is:

$d e t (H_{f}) = (\frac{\partial^{2} f}{\partial x^{2}}) (\frac{\partial^{2} f}{\partial y^{2}}) - {(\frac{\partial^{2} f}{\partial x \partial y})}^{2} = 6 \times 12 - 0 = 72$

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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.
The Hessian and discriminant of a function $f (x, y)$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Defining hessian and discriminant

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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.The Hessian and discriminant of a functionf(x,y).

Short Answer

Step by step solution

Step 1. Given information

Step 2. Defining hessian and discriminant

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Mechanics Maths

Discrete Mathematics

Logic and Functions

Pure Maths

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.
The Hessian and discriminant of a function $f (x, y)$ .