# The valuesofcisthereastraightlinethatintersectsthecurveinfourdistinctpoints.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 4, Problem 16P
To determine

## To find: The valuesofcisthereastraightlinethatintersectsthecurveinfourdistinctpoints.

Expert Solution

The values of care 10.164<c<6.2253orc<6.7894

### Explanation of Solution

Given information:

The givencurve is,

y=x4+cx3+12x25x+2

Concept used:

The distinctformulais

D=a2b24b34a3c27c2+18abc

The givenequationis,

y=x4+cx3+12x25x+2....(1.)

Todifferentiatetheequationwithrespectto x

y'=4x3+3cx2+24x5=x3+34cx2+6x54

Comparetheequation (1.) withcubicequationis

x3+ax2+bx+c=0

a=3c4,b=6,c=54

Thedistinctis,

D=a2b24b34a3c27c2+18abc=(3c4)2624(6)34(3c4)3(54)27(54)2+18(3c4)(6)(54)=20.25c2864+2.109375c342.1875101.25c=2.109375c3+20.25c2101.25c906.1875

Bysolvingthe equation therearethreerootsare

x1=6.7194x2=10.1641x3=6.2253

Therefore, the value of c is 10.164<c<6.2253orc<6.7894

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