/*! 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 3 The least value of the cxpressio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 3

The least value of the cxpression \(x^{2}+4 y^{2}+3 z^{2}-2 x\) \(-12 y-6 z+14 i s\) a. 1 b. no least value c. 0 d. none of these

Short Answer

Expert verified
1

Step by step solution

01

Rewrite the expression

Begin by taking the initial expression given: \[f(x, y, z) = x^2 + 4y^2 + 3z^2 - 2x - 12y - 6z + 14\]Our goal is to find the minimum value of this expression.
02

Complete the square for x

Focus on the terms involving x: \[x^2 - 2x\]Complete the square:\[(x^2 - 2x + 1 -1) = (x - 1)^2 - 1\]
03

Complete the square for y

Now focus on the terms involving y:\[4y^2 - 12y\]Factor out the 4:\[4(y^2 - 3y)\]Complete the square inside the parentheses:\[4((y^2 - 3y + \,\left(\frac{3}{2}\right)^2 - \,\left(\frac{3}{2}\right)^2) = 4((y - \,\frac{3}{2})^2 - \,\frac{9}{4}) = 4(y - \,\frac{3}{2})^2 - 9\]
04

Complete the square for z

Focus on the terms involving z:\[3z^2 - 6z\]Factor out the 3:\[3(z^2 - 2z)\]Complete the square inside the parentheses:\[3((z^2 - 2z + 1 - 1) = 3((z - 1)^2 - 1) = 3(z - 1)^2 - 3\]
05

Combine the completed squares

Substitute these completed squares back into the expression to rewrite it:\[f(x, y, z) = (x - 1)^2 + 4(y - \,\frac{3}{2})^2 + 3(z - 1)^2 - 1 - 9 - 3 + 14\]Simplify the constants:\[-1 - 9 - 3 + 14 = 1\]Thus the expression becomes:\[f(x, y, z) = (x - 1)^2 + 4(y - \,\frac{3}{2})^2 + 3(z - 1)^2 + 1\]
06

Determine the minimum value

The values \((x - 1)^2\), \(4(y - \,\frac{3}{2})^2\), and \(3(z - 1)^2\) are always non-negative because they are squares. The minimum of the sum of these terms is zero, occurring when each term is zero:\[(x-1) = 0, \, (y-\frac{3}{2}) = 0, \, \text{and} \, (z-1) = 0\]Thus, \(x = 1, y = \frac{3}{2}, \text{and} \, z = 1\). The minimum value of the expression is reached when these conditions are met, resulting in \(f(x, y, z) = 1\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Completing the Square
Completing the square is a valuable technique used to simplify quadratic expressions. It allows us to rewrite expressions in a form that makes it easier to identify their properties, such as vertex form for parabolas. In our exercise, we have the expression \(x^2 - 2x\). To complete the square:
  • Consider the term \(-2x\) and find half of the coefficient of \(x\), which is \(-1\).
  • Square \(-1\) to get \(1\).
  • Rewrite \(x^2 - 2x + 1\) as \((x - 1)^2\), and subtract \(1\) because it was added unnecessarily, resulting in \((x-1)^2 - 1\).
Similarly, complete the square for variables \(y\) and \(z\) in the expression. This process transforms a quadratic into a squared binomial, making it clear how the variable contributes to the overall minimum or maximum of the expression.
Non-negative Values
In mathematics, understanding non-negative values is crucial, especially in the context of squares. Non-negative numbers are those that are greater than or equal to zero. When completing the square, each term results in a squared binomial, such as \((x - 1)^2\). Since any real number squared is non-negative:
  • \((x - 1)^2\), \(4(y - \frac{3}{2})^2\), and \(3(z - 1)^2\) produce non-negative results.
  • This implies that no matter the values of \(x\), \(y\), or \(z\), these portions of the expression will never be negative.
  • Ensuring understanding of these concepts is key to minimizing quadratic expressions successfully.
Non-negative values are foundational in determining minima and maxima, as squaring any deviation from a point only increases, grounding the search for minimal values to when the squared term itself is zero.
Quadratic Expression
A quadratic expression involves the square of variable terms and is typically in the form \(ax^2 + bx + c\). They feature prominently in various mathematical settings, such as physics and engineering. In this exercise, the expression \(x^2 + 4y^2 + 3z^2 - 2x - 12y - 6z + 14\) is a multivariable quadratic:
  • The term \(x^2\) is a basic quadratic term for \(x\).
  • Coefficients like \(4\) in \(4y^2\) or \(3\) in \(3z^2\) indicate the contribution of each variable.
  • Applying completing-the-square allows restructuring into sums or differences of squares, providing a route to the minimum value determination.
Recognizing the structure of quadratic forms and using algebraic techniques like completing the square renders these expressions more manageable and informative.
Minimum Value Determination
To find the minimum value of an expression, especially a quadratic one, one effective strategy is to rely on its squared components. With the expression reformulated into completed squares, it becomes straightforward:
  • The expression \((x-1)^2 + 4(y - \frac{3}{2})^2 + 3(z-1)^2 + 1\) is in a form where each squared term reaches its minimum value of zero.
  • This occurs when \(x = 1\), \(y = \frac{3}{2}\), and \(z = 1\).
  • The overall minimum of the expression is the constant term left, which is \(1\).
This strategy highlights the power of algebraic manipulation to reveal the role that the constant term plays as the expression鈥檚 minimal value, simplifying the evaluation to conditions that each variable zeroes its respective completed-square component.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The range of \(a\) for which the equation \(x^{2}+a x-4=0\) has its smailer root in the interval \((-1,2)\) is a. \((-\infty,-3)\) b. \((0,3)\)C. \((0, \infty)\) d. \((-\infty,-3) \cup(0, \infty)\)

A quadratic equation whose product of roots \(x_{1}\) and \(x_{2}\) is equal to 4 and satisfying the rclation \(x_{1} /\left(x_{1}-1\right)+x_{2} /\left(x_{2}-1\right)=2\) is a. \(x^{2}-2 x+4=0\) b. \(x^{2}+2 x+4=0\) c. \(x^{2}+4 x+4=0\) d. \(x^{2}-4 x+4=0\)

If \(x \in R\), then prove thatmaximumvalue of \(2(a-x)\left(x+\sqrt{x^{2}+b^{2}}\right)\) is \(a^{2}+b^{2}\)

The integral values of \(m\) for which the roos of the equation \(m x^{2}+(2 m-1) x+(m-2)=0\) are rational are given by the expression [where \(n\) is integer] a. \(n^{2}\) b. \(n(n+2)\) c. \(n(n+1)\) d. none of these

The number of irrational roots of the equation \(4 x /\left(x^{2}+x+3\right)\) \(+5 x /\left(x^{2}-5 x+3\right)=-3 / 2\) is a. 4 b. 0 c. 1 d. 2
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.