Da Nang-08/2014

Natural Science Department – Duy Tan University

Lecturer: Ho Xuan Binh

MAXIMUM & MINIMUM

VALUES In this section, we will learn:

How to use partial derivatives to locate

maxima and minima of functions of two variables

MAXIMUM & MINIMUM VALUES 1 Natural Science Department – Duy Tan University

Maximum and minimum values 

Definition 2Natural Science Department – Duy Tan University

A function of two variables has a local maximum at (a, b) if f(x, y) ≤ f(a, b) when (x, y) is near (a, b).This means that f(x, y) ≤ f(a, b) for all points (x, y) in some disk with center (a, b). The number f(a, b) is called a local maximum value.

Maximum and minimum values 

If f(x, y) ≥ f(a, b) when (x, y) is near (a, b), then f has a local minimum at (a, b). The number f(a, b) is a local minimum value.

Theorem 13Natural Science Department – Duy Tan University

Maximum and minimum values 

If f has a local maximum or minimum at (a, b) and the first-order partial derivatives of f exist there, then

fx(a, b) = 0 and fy(a, b) = 0

CRITICAL POINT4Natural Science Department – Duy Tan University

A point (a, b) is called a critical point (or stationary point) of f if either:

Maximum and minimum values 

* fx(a, b) = 0 and fy(a, b) = 0

* One of these partial derivatives does not exist.

SADDLE POINT5

Natural Science Department – Duy Tan University

Near the origin, the graph has the shape of a saddle.

So, (0, 0) is called a saddle point of f(x,y) = y2 – x2.

Maximum and minimum values 

Theorem 2 6Natural Science Department – Duy Tan University

Suppose that:

Maximum and minimum values 

The second partial derivatives of f are continuous on a disk with center (a, b).

fx(a, b) = 0 and fy(a, b) = 0 [that is, (a, b) is a critical point of f].

Let D = D(a, b) = fxx(a, b) fyy(a, b) – [fxy(a, b)]2

Theorem 2 6Natural Science Department – Duy Tan University

Maximum and minimum values 

b. If D > 0 and fxx(a, b) < 0, f(a, b) is a local maximum.

a. If D > 0 and fxx(a, b) > 0, f(a, b) is a local minimum.

c. If D < 0, f(a, b) is not a local maximum or minimum.

* In case c, The point (a, b) is called a saddle point of f .

* If D = 0, the test gives no information: f could have a local maximum or local minimum at (a,b), or (a,b) could be a saddle point of f.

Notes:

Example 7Natural Science Department – Duy Tan University

Find the local maximum and minimum values and saddle points of

Maximum and minimum values 

f(x, y) = x4 + y4 – 4xy + 1

LOGO

Thank you for your attention