/*! 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 1 In each of Exercises \(1-6,\) a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 1

In each of Exercises \(1-6,\) a function \(f\) is given. Locate each point \(c\) for which \(f(c)\) is a local extremum for \(f\). (Calculus is not needed for these exercises.) $$ f(x)=(x-2)^{2}+3 $$

Short Answer

Expert verified
The point \((2, 3)\) is a local minimum for \(f(x)\).

Step by step solution

01

Understand the function form

The given function is defined as \( f(x) = (x-2)^2 + 3 \). This is a quadratic function, which represents a parabola opening upwards because the coefficient of \( (x-2)^2 \) is positive.
02

Identify the vertex

For the quadratic function \( f(x) = a(x-h)^2 + k \), the vertex is located at the point \((h, k)\). In this function, \( a = 1 \), \( h = 2 \), and \( k = 3 \). Therefore, the vertex is at the point \((2, 3)\).
03

Determine local extrema points

Since the parabola opens upwards, the vertex at \((2, 3)\) is the lowest point. Hence, this point represents the local minimum of the function \( f(x) \). There are no local maxima for this parabola since it opens upwards.

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.

Understanding Quadratic Functions
Quadratic functions are a type of polynomial function where the highest degree of the variable is 2. They are generally written in the standard form as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). In the context of our exercise, the quadratic function given is \( f(x) = (x-2)^2 + 3 \). Here, the function is expressed in a slightly different form: \( a(x-h)^2 + k \), known as the vertex form. Each part of this function tells us important information:
  • \( a \): describes the direction of the parabola. If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
  • \((h, k)\): gives the coordinates of the vertex of the parabola.
This function simplifies our work in identifying the parabola's crucial attributes, like its vertex and direction.
Vertex of a Parabola
The vertex of a parabola is the highest or lowest point on the graph, depending on the orientation of the parabola. For the function \( f(x) = (x-2)^2 + 3 \), the vertex can be easily identified thanks to its vertex form, \( a(x-h)^2 + k \).
  • Here, \( h = 2 \) and \( k = 3 \).
  • The vertex is the point \((h, k)\), so it's \((2, 3)\) for our function.
This vertex at \((2, 3)\) represents a single point that signifies not only the turn of the parabola but also the minimum value the function can achieve in the context of an upward-opening parabola.
What is a Local Minimum?
A local minimum is the lowest point in a particular section of a graph of a function, meaning there are no values less than it in its immediate vicinity. In our exercise, we explored \( f(x) = (x-2)^2 + 3 \). The graph of this function is a parabola that opens upwards. This tells us:
  • Since the parabola opens upwards, the vertex at \((2, 3)\) is the lowest point.
  • This point, \((2, 3)\), is where the local minimum occurs.
There is no local maximum in this case, because an upward-opening parabola only dips down to its lowest point before rising indefinitely on either side. Understanding the concept of local minimum helps in identifying key points on graphs and optimizing solutions in real-world problems.

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

A model rocket is launched straight up with an initial velocity of \(100 \mathrm{ft} / \mathrm{s} .\) The fuel on board the rocket lasts for 8 seconds and maintains the rocket at this upward velocity (countering the negative acceleration due to gravity). After the fuel is spent then gravity takes over. What is the greatest height that the rocket reaches? With what velocity does the rocket strike the ground?

Use trigonometric identities to compute the indefinite integrals. \(\text { Evaluate } \int \frac{1}{2^{x}} d x\)

If \(T\) is the temperature in degrees Kelvin of a white dwarf star, then the rate at which \(T\) changes as a function of time \(t\) is given by \(d T / d t=-\alpha T^{7 / 2}\) where \(\alpha\) is a positive constant. Do white dwarfs cool or heat up? Is the rate at which a \(\begin{array}{lllll}\text { white } & \text { dwarf's } & \text { temperature } & \text { changes } & \text { increasing } & \text { or }\end{array}\) decreasing? Explain your answers.

The initial temperature \(T(0)\) of an object is \(50^{\circ} \mathrm{C}\). If the object is cooling at the rate $$T^{\prime}(t)=-0.2 e^{-0.02 t}$$ when measured in degrees centigrade per second, what is the limit \(T_{\infty}\) of its temperature as \(t\) tends to infinity? How much time \(\tau_{1}\) does it take for the object to \(\operatorname{cool} 4^{\circ} \mathrm{C}\) (from \(50^{\circ} \mathrm{C}\) to \(\left.46^{\circ} \mathrm{C}\right) ?\) How much additional time \(\tau_{2}\) elapses before the object cools an additional \(2^{\circ} \mathrm{C}\) ? How much additional time \(\tau_{3}\) elapses before the object cools an additional \(1^{\circ} \mathrm{C}\) (from \(44^{\circ} \mathrm{C}\) to \(\left.43^{\circ} \mathrm{C}\right)\) ?

In each of Exercises 54-60, determine each point \(c,\) where the given function \(f\) satisfies \(f^{\prime}(c)=0\). At each such point, use the First Derivative Test to determine whether \(f\) has a local maximum, a local minimum, or neither. $$ f(x)=\tanh ^{-1}(\sqrt{x})-2 \sqrt{x} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

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.