/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 What is the procedure for locati... [FREE SOLUTION] | 91影视

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Chapter 12: 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: Find the absolute maximum and minimum values of the function f(x) = x鲁 - 6x虏 + 9x + 2 on the closed interval [0, 6]. Solution: Step 1: Find the critical points We first find the derivative of the function: f'(x) = 3x虏 - 12x + 9. To find the critical points, set f'(x) = 0. 3x虏 - 12x + 9 = 0 x虏 - 4x + 3 = 0 (x - 1)(x - 3) = 0 x = 1 and x = 3 are the critical points. Step 2: Evaluate the function at the critical points and endpoints Now, evaluate the function at the critical points (x = 1 and x = 3) and endpoints (x = 0 and x = 6). f(0) = 0鲁 - 6(0)虏 + 9(0) + 2 = 2 f(1) = 1鲁 - 6(1)虏 + 9(1) + 2 = 6 f(3) = 3鲁 - 6(3)虏 + 9(3) + 2 = 2 f(6) = 6鲁 - 6(6)虏 + 9(6) + 2 = 38 Step 3: Identify the absolute maximum and minimum values From the table in Step 2, we can see that the highest function value is 38, which occurs at x = 6. The lowest function value is 2, which occurs at x = 0 and x = 3. Therefore, the absolute maximum value is 38 at x = 6, and the absolute minimum value is 2 at x = 0 and x = 3.

Step by step solution

01

Find the critical points

To find the critical points, take the derivative of the function and set it equal to zero. Solve the equation to find the critical points. These points are candidates for the locations of the absolute maximum and minimum values.
02

Evaluate the function at the critical points and endpoints

Plug in the critical points and endpoints of the closed bounded domain into the original function to find the function values at these points. Create a table and list the points along with their corresponding function values.
03

Identify the absolute maximum and minimum values

Compare the function values from Step 2. The highest value will be the absolute maximum value, and the lowest value will be the absolute minimum value. The points at which these values occur are the locations of the absolute maximum and minimum values on the closed bounded domain.

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Most popular questions from this chapter

In the advanced subject of complex variables, a function typically has the form \(f(x, y)=u(x, y)+i v(x, y),\) where \(u\) and \(v\) are real-valued functions and \(i=\sqrt{-1}\) is the imaginary unit. A function \(f=u+i v\) is said to be analytic (analogous to differentiable) if it satisfies the Cauchy-Riemann equations: \(u_{x}=v_{y}\) and \(u_{y}=-v_{x}\). a. Show that \(f(x, y)=\left(x^{2}-y^{2}\right)+i(2 x y)\) is analytic. b. Show that \(f(x, y)=x\left(x^{2}-3 y^{2}\right)+i y\left(3 x^{2}-y^{2}\right)\) is analytic. c. Show that if \(f=u+i v\) is analytic, then \(u_{x x}+u_{y y}=0\) and \(v_{x x}+v_{y y}=0 .\) Assume \(u\) and \(v\) satisfy the conditions in Theorem 12.4.

Let \(x, y,\) and \(z\) be non-negative numbers with \(x+y+z=200\) a. Find the values of \(x, y,\) and \(z\) that minimize \(x^{2}+y^{2}+z^{2}\) b. Find the values of \(x, y,\) and \(z\) that minimize \(\sqrt{x^{2}+y^{2}+z^{2}}\). c. Find the values of \(x, y,\) and \(z\) that maximize \(x y z\) d. Find the values of \(x, y,\) and \(z\) that maximize \(x^{2} y^{2} z^{2}\).

The angle between two planes is the angle \(\theta\) between the normal vectors of the planes, where the directions of the normal vectors are chosen so that \(0 \leq \theta<\pi\) Find the angle between the planes \(5 x+2 y-z=0\) and \(-3 x+y+2 z=0\)

Consider the following functions \(f\). a. Is \(f\) continuous at (0,0)\(?\) b. Is \(f\) differentiable at (0,0)\(?\) c. If possible, evaluate \(f_{x}(0,0)\) and \(f_{y}(0,0)\). d. Determine whether \(f_{x}\) and \(f_{y}\) are continuous at (0,0). e. Explain why Theorems 12.5 and 12.6 are consistent with the results in parts \((a)-(d)\). $$f(x, y)=\sqrt{|x y|}$$

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the x y-, x z^{-}, \text {and } y z-\text {traces, when they exist. c. Sketch a graph of the surface. $$-\frac{x^{2}}{6}-24 y^{2}+\frac{z^{2}}{24}-6=0$$
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