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Chapter 7: Problem 9

Find the relative extreme values of each function. $$f(x, y)=3 x-2 y-6$$

Short Answer

Expert verified
The function has no relative extreme values; it is linear.

Step by step solution

01

Identify the Critical Points

Since the given function is a linear function, it's important to recognize that it has no relative extreme values because its gradient is constant throughout the domain. In other words, critical points do not exist for this function.
02

Understand the Nature of the Function

A linear function like this does not exhibit either local minima or maxima because it represents a plane with a uniform slope and extends infinitely without any curvature. Thus, it cannot have relative extreme values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Functions
In multivariable calculus, linear functions are a fundamental concept. They are functions of the form \( f(x, y) = ax + by + c \), where \( a \), \( b \), and \( c \) are constants. Here, our function is \( f(x, y)=3x-2y-6 \). Linear functions describe planes in three-dimensional space, reflecting a direct proportional relationship between the variables.
Characteristics of linear functions include:
  • Constant gradients, meaning that the rate of change is uniform across the plane.
  • They do not bend or curve; instead, the surface they form is flat.
  • No peaks or valleys, indicating no significant changes in direction.
This uniformity makes them straightforward yet powerful, particularly in scenarios requiring constant rates of change.
Critical Points
Critical points in multivariable calculus are points where a function's gradient is zero or undefined. They are crucial because they help identify potential points of relative minima, maxima, or saddle points.
For a function \( f(x, y) \):
  • Compute the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).
  • Set these derivatives to zero and solve for \( x \) and \( y \) to find critical points.
However, for linear functions like \( f(x, y)=3x-2y-6 \), the partial derivatives are constants, \( \frac{\partial f}{\partial x} = 3 \) and \( \frac{\partial f}{\partial y} = -2 \). This constancy indicates no critical points, because the gradient never zeroes out. Hence, there are no relative extreme values present.
Gradient
The gradient of a function is a vector that indicates the direction of the steepest ascent at any given point. For a function \( f(x, y) \), the gradient is the vector composed of its partial derivatives: \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \).
Key points about gradients:
  • The magnitude of the gradient specifies how steep the slope is in that direction.
  • The direction vector points towards the greatest rate of increase.
  • For linear functions, the gradient is constant throughout the domain.
With \( f(x, y)=3x-2y-6 \), the gradient \( abla f = (3, -2) \). This means that the function increases by 3 units in the \( x \) direction and decreases by 2 units in the \( y \) direction consistently. This flat, uniform character of the gradient aligns with the absence of critical points.

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