/*! 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 31 Give an example of a nonconstant... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 31

Give an example of a nonconstant function \(f:(-1,1) \rightarrow \mathbb{R}\) such that \(f\) has a local extremum at 0, and 0 is a point of inflection for \(f\).

Short Answer

Expert verified
An example of a nonconstant function \(f(x)\) defined on the interval (-1, 1) with a local extremum and a point of inflection at x = 0 is \(f(x) = x^4\).

Step by step solution

01

Choose an appropriate function

Let's consider the polynomial function \(f(x) = x^4\). This function is non-constant and defined on the open interval (-1,1).
02

Calculate the first and second derivatives of f

We'll now verify if f'(0) = 0 and f''(0) = 0. 1. Calculate the first derivative of f(x): \(f'(x) = \frac{d}{dx}(x^4) = 4x^3\) 2. Calculate the second derivative of f(x): \(f''(x) = \frac{d^2}{dx^2}(x^4) = \frac{d}{dx}(4x^3) = 12x^2\)
03

Check if f'(0) = 0 and f''(0) = 0

Evaluate f'(0) and f''(0) using the derivatives we found in Step 2: 1. \(f'(0) = 4(0)^3 = 0\) 2. \(f''(0) = 12(0)^2 = 0\) As we can see, f'(0) = 0 and f''(0) = 0, so 0 is both a local extremum and a point of inflection for f(x) = x^4. Therefore, the function \(f(x) = x^4\) is an example of a nonconstant function defined on the interval (-1, 1) with a local extremum and a point of inflection at x = 0.

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!

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

Prove that every rational number \(r\) can be written as \(r=\frac{p}{q}\), where \(p, q \in \mathbb{Z}, q>0\) and \(p, q\) are relatively prime, and moreover the integers \(p\) and \(q\) are uniquely determined by \(r .\)

Use the Cauchy-Schwarz inequality to prove the A.M.-H.M. inequality, namely, if \(n \in \mathbb{N}\) and \(a_{1}, \ldots, a_{n}\) are positive real numbers, then prove that $$ \frac{a_{1}+\cdots+a_{n}}{n} \geq \frac{n}{r}, \quad \text { where } \quad r:=\frac{1}{a_{1}}+\cdots+\frac{1}{a_{n}} . $$

Given any function \(f: \mathbb{R} \rightarrow \mathbb{R}\), prove the following: (i) If \(f(x y)=f(x)+f(y)\) for all \(x, y \in \mathbb{R}\), then \(f(x)=0\) for all \(x \in \mathbb{R}\). [Note: It is, however, possible that there are nonzero functions defined on subsets of \(\mathbb{R}\), such as \((0, \infty)\) that satisfy \(f(x y)=f(x)+f(y)\) for all \(x, y\) in the domain of \(f\). A prominent example of this is the logarithmic function, which will be discussed in Section 7.1.] (ii) If \(f(x y)=f(x) f(y)\) for all \(x, y \in \mathbb{R}\), then either \(f(x)=0\) for all \(x \in \mathbb{R}\), or \(f(x)=1\) for all \(x \in \mathbb{R}\), or \(f(0)=0\) and \(f(1)=1\). Further, \(f\) is either an even function or an odd function. Give examples of even as well as odd functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) satisfying \(f(x y)=f(x) f(y)\) for all \(x, y \in \mathbb{R}\)

Let \(p(x), q(x) \in \mathbb{R}[x]\). Show that \(p(x)\) and \(q(x)\) are relatively prime if and only if \(u(x) p(x)+v(x) q(x)=1\) for some \(u(x), v(x) \in \mathbb{R}[x] .\) Is it true that if a nonzero polynomial \(d(x) \in \mathbb{R}[x]\) satisfies \(u(x) p(x)+v(x) q(x)=d(x)\) for some \(u(x), v(x) \in \mathbb{R}[x]\), then \(d(x)\) is a GCD of \(p(x)\) and \(q(x)\) ?

Show that a nonzero polynomial in \(\mathbb{R}[x]\) of degree \(n\) has at most \(n\) roots in \(\mathbb{R}\). (Hint: Exercise 59.)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.