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Chapter 13: Problem 8

What is the procedure for locating absolute maximum and minimum values on a closed bounded domain?

Short Answer

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Question: Describe the procedure for finding absolute maximum and minimum values on a closed bounded domain. Answer: The procedure for finding absolute maximum and minimum values on a closed bounded domain involves the following steps: 1) Understand the concept of absolute maxima and minima. 2) Identify critical points by solving the derivative of the function equal to zero or where it's undefined. 3) Evaluate the function at critical points and endpoints of the interval. 4) Compare the function values to identify absolute maximum and minimum values. Always ensure that the domain is closed and bounded before proceeding.

Step by step solution

01

Understand Absolute Maxima and Minima

Absolute maximum and minimum values are the highest and lowest function values on a specified domain, respectively. The location of these maximum and minimum values can be found using calculus methods, specifically by identifying critical points.
02

Identify Critical Points

Critical points are where the function has a relative maximum or minimum, and are found by checking where the derivative of the function (f'(x)) is zero or undefined. In other words, find critical points by solving f'(x) = 0.
03

Apply Procedure for Finding Absolute Maximum and Minimum

To find the absolute maximum and minimum values, follow these steps: 1. Determine the function's critical points by finding where its derivative is equal to zero or undefined. 2. Evaluate the function at each critical point and at the endpoints of the closed interval. 3. Identify the absolute maximum and minimum values by comparing the function values obtained in the previous step.
04

Verify if the Domain is Closed and Bounded

A domain is closed if it includes its boundary points and bounded if it is defined over a finite interval. Make sure that the given function is defined on a closed and bounded domain before applying the previously mentioned procedure. In summary, to find the absolute maximum and minimum values of a function on a closed bounded domain, we must locate the critical points, evaluate the function at these points and the endpoints, and compare the results to determine the highest and lowest values. Always verify that the domain is closed and bounded before proceeding.

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Most popular questions from this chapter

A function of one variable has the property that a local maximum (or minimum) occurring at the only critical point is also the absolute maximum (or minimum) (for example, \(f(x)=x^{2}\) ). Does the same result hold for a function of two variables? Show that the following functions have the property that they have a single local maximum (or minimum), occurring at the only critical point, but that the local maximum (or minimum) is not an absolute maximum (or minimum) on \(\mathbb{R}^{2}\). a. \(f(x, y)=3 x e^{y}-x^{3}-e^{3 y}\) b. \(f(x, y)=\left(2 y^{2}-y^{4}\right)\left(e^{x}+\frac{1}{1+x^{2}}\right)-\frac{1}{1+x^{2}}\) This property has the following interpretation. Suppose that a surface has a single local minimum that is not the absolute minimum. Then water can be poured into the basin around the local minimum and the surface never overflows, even though there are points on the surface below the local minimum.

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(0,0)} \frac{|x-y|}{|x+y|}$$

Use the Second Derivative Test to prove that if \((a, b)\) is a critical point of \(f\) at which \(f_{x}(a, b)=f_{y}(a, b)=0\) and \(f_{x x}(a, b)<0

Prove that for the plane described by \(f(x, y)=A x+B y,\) where \(A\) and \(B\) are nonzero constants, the gradient is constant (independent of \((x, y)\) ). Interpret this result.

Let \(R\) be a closed bounded set in \(\mathbb{R}^{2}\) and let \(f(x, y)=a x+b y+c,\) where \(a, b,\) and \(c\) are real numbers, with \(a\) and \(b\) not both zero. Give a geometrical argument explaining why the absolute maximum and minimum values of \(f\) over \(R\) occur on the boundaries of \(R\)
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