/*! 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 9 Sketch the graph of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 9

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=3 x^{4}+4 x^{3}\)

Short Answer

Expert verified
The graph peaks at x=0 and x=-1, with points of inflection at x=0 and x=-2/3

Step by step solution

01

Find the First Derivative

First find the derivative (\(y'\)) of the function \(y=3x^{4}+4x^{3}\) using the power rule for differentiation, which states that the derivative of \(x^n\) is \(nx^{n-1}\). The first derivative of the function is: \(y'=12x^{3}+12x^{2}\)
02

Find the Critical Points

The critical points occur where the first derivative (\(y'\)) is equal to zero or undefined. For this function, the derivative is never undefined. So, set the derivative equal to zero and solve for \(x\): \(0=12x^{3}+12x^{2}\) --> \(0=x^{2}(12x+12)\). Therefore the critical points are \(x=0\) and \(x=-1\)
03

Find the Second Derivative

Differentiate the equation for the first derivative once more using the power rule to get the second derivative (\(y''\)): \(y''=36x^{2}+24x\)
04

Find the Points of Inflection

Set the second derivative equal to zero and solve for \(x\) to find the points of inflection: \(0=36x^{2}+24x\) --> \(0=12x(3x+2)\). Therefore the points of inflection are \(x=0\) and \(x=-2/3\).
05

Sketch the Graph

Use the critical points and points of inflection to sketch the graph. Label these points on the graph and pay attention to how the behavior of the function changes at these points. For \(x>0\), the function is decreasing and concave up. For \(-1<x<0\) the function is increasing and concave up, while for \(x<-1\) it's increasing and concave down. Recognize the symmetry of derivative sign changes around the points of inflection.

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

The cost and revenue functions for a product are \(C=34.5 x+15,000\) and \(R=69.9 x\) (a) Find the average profit function \(\bar{P}=(R-C) / x\). (b) Find the average profits when \(x\) is \(1000,10,000\), and 100,000 (c) What is the limit of the average profit function as \(x\) approaches infinity? Explain your reasoning.

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{4}-4 x^{3}+16 x\)

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=\frac{x^{2}}{x^{2}+3}\)

Find the differential \(d y\). \(y=(4 x-1)^{3}\)

Find the differential \(d y\). \(y=\frac{x+1}{2 x-1}\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Decision 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.