/*! 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 74 True or False: At a critical num... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 74

True or False: At a critical number the function must be defined.

Short Answer

Expert verified
True, the function must be defined at a critical number.

Step by step solution

01

Understand the Definition of a Critical Number

A critical number of a function \( f(x) \) is a number \( c \) in the domain of \( f(x) \) such that either the derivative \( f'(c) = 0 \) or the derivative \( f'(c) \) does not exist. This means that a critical point is where the function can potentially have a maximum, minimum, or a saddle point.
02

Verify the Domain Requirement

A critical number must be in the domain of the function. Since the function must be defined at \( c \), it emphasizes that \( c \) must belong to the set of all possible inputs of the function (i.e., its domain). If a function is not defined at \( c \), \( f(c) \) does not exist, and thus \( c \) cannot be a critical number.
03

Analyze the Statement

The statement "At a critical number the function must be defined" implies that a function must have a value at this point to consider it a critical number. Based on the definition from Step 1 and the requirement from Step 2, the statement is True.

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.

Critical Number
A critical number is an essential concept in calculus that helps us understand the behavior of a function at specific points. These points are where the function's derivative either equals zero or does not exist. Here鈥檚 why these points are crucial:
  • If the derivative equals zero, the slope of the tangent line to the graph of the function is horizontal, which may indicate a local maximum, minimum, or a saddle point.
  • If the derivative does not exist, it might indicate a sharp turn or cusp in the graph of the function.
For a critical number to be valid, the function must be defined at that point. This means that the critical number must lie within the function's domain. Without the function being defined at those particular points, we cannot specify them as critical numbers.
Domain of a Function
The domain of a function is the complete set of all possible input values (often represented as 'x') for which the function is defined. This is a crucial concept to understand when identifying critical numbers. Why so?
  • Critical numbers must be within this set because if a function is not defined at a point, you cannot find a derivative there, nor can you evaluate the function's behavior.
  • The domain dictates where you can and cannot find critical points, which influence how you analyze the function for extrema (maximums and minimums) and inflection points.
Understanding the domain is thus fundamental in calculus as it dictates the scope within which you solve problems involving derivatives or limits.
Derivative
The derivative of a function gives us vital information about the function's rate of change at any given point. It is denoted as \( f'(x) \) for a function \( f(x) \). Here's why derivatives matter for finding critical numbers:
  • The derivative equals zero \((f'(c) = 0)\) at critical numbers, indicating possible local maxima or minima.
  • When the derivative does not exist at a point within the domain, it also could signify a critical number, often related to sharp turns or unpredictable changes in the graph.
To determine a function's critical points, you have to find where \( f'(x) = 0 \) or \( f'(x) \) is undefined but the function \( f(x) \) itself is defined, and these points help us make sense of the graph's overall behavior.

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

GENERAL: Athletic Fields A running track consists of a rectangle with a semicircle at each end, as shown below. If the perimeter is to be exactly 440 yards, find the dimensions \((x\) and \(r)\) that maximize the area of the rectangle. [Hint: The perimeter is \(2 x+2 \pi r .]\)

If the amount of a drug in a person's blood after \(t\) hours is \(f(t)=t /\left(t^{2}+9\right)\), when will the drug concentration be the greatest?

For each demand equation, use implicit differentiation to find \(d p / d x\). $$ x p^{2}=96 $$

If a government wants to increase revenue, why not just increase the tax rate? Explain.

A case of vintage wine appreciates in value each year, but there is also an annual storage charge. The value of a typical case of investment-grade wine after \(t\) years is \(V(t)=2000+80 \sqrt{t}-10 t\) dollars (for \(0 \leq t \leq 25\) ). Find the storage time that will maximize the value of the wine.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics 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.