/*! 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 11 Sketch the graph of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 11

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{3}-6 x^{2}+3 x+10\)

Short Answer

Expert verified
The critical points are determined by solving \(3x^{2}-12x+3=0\) and the points of inflection by solving \(6x-12=0\). With these points along with the origin and y-intercept marked, one can sketch the function.

Step by step solution

01

Find the derivative

First find the derivative of the given function. The derivative of \(y=x^{3}-6x^{2}+3x+10\) is \(y'=3x^{2}-12x+3\).
02

Set the derivative equal to zero

To find the critical points, set the derivative equal to zero, i.e solve the equation \(3x^{2}-12x+3=0\).
03

Calculate the second derivative

To find the points of inflection and relative extrema, calculate the second derivative. The second derivative of the given function is \(y''=6x-12\).
04

Set the second derivative equal to zero

Set the second derivative equal to zero, and solve for \(x\) to find the points of inflection. ie. \(6x-12=0\)
05

Sketch the graph

Now it's time to sketch the graph. On the y-axis, mark the points where the function crosses it (origin and y-intercept). Mark the critical points and the points of inflection on the x-axis. Also mark any relative extrema, marking whether they're local maximums or minimums. You have now marked all the necessary points which will help you sketch your graph.

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 \(C=25.5 x+1000\) and \(R=75.5 x\) (a) Find the average profit function \(\bar{P}=\frac{R-C}{x}\). (b) Find the average profits when \(x\) is 100,500 , and 1000 . (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. \(y=3 x^{3}-9 x+1\)

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=2 x^{2}-4 x+1\)

The body surface area (BSA) of a 180-centimeter-tall (about six-feet-tall) person is modeled by $$ B=0.1 \sqrt{5 w} $$ where \(B\) is the BSA (in square meters) and \(w\) is the weight (in kilograms). Use differentials to approximate the change in the person's BSA when the person's weight changes from 90 kilograms to 95 kilograms.

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=\frac{x^{4}}{x^{4}-1}\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.