Find all the local maxima, local minima, and saddle points of the functions in Exercises 1–4. 1. ƒ(x, y) = 2x2 + 3xy + 4y2 - 5x + 2y 2. ƒ(x, y) = x2 - 2xy + 2y2 - 2x + 2y + 1 3. ƒ(x, y) = x2 - y2 - 2x + 4y + 6 4. ƒ(x, y) = x2 + 2xy

Find all the local maxima, local minima, and saddle points of the functions in Exercises 1–4. 1. ƒ(x, y) = 2x2 + 3xy + 4y2 - 5x + 2y 2. ƒ(x, y) = x2 - 2xy + 2y2 - 2x + 2y + 1 3. ƒ(x, y) = x2 - y2 - 2x + 4y + 6 4. ƒ(x, y) = x2 + 2xy

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.5: Applications

Problem 2EQ

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Question

Find all the local maxima, local minima, and saddle points of the functions in Exercises 1–4.

1. ƒ(x, y) = 2x2 + 3xy + 4y2 - 5x + 2y

2. ƒ(x, y) = x2 - 2xy + 2y2 - 2x + 2y + 1

3. ƒ(x, y) = x2 - y2 - 2x + 4y + 6

4. ƒ(x, y) = x2 + 2xy

Formula Formula A function f ( x ) is also said to have attained a local minimum at x = a , if there exists a neighborhood ( a − δ , a + δ ) of a such that, f ( x ) > f ( a ) , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a f ( x ) − f ( a ) > 0 , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a In such a case f ( a ) is called the local minimum value of f ( x ) at x = a .

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