   Chapter 4.7, Problem 56E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# At which points on the curve y = 1 + 40x3 − 3x5 does the tangent line have the largest slope?

To determine

To find: The points on the curve y=1+40x33x5 such that the tangent line on that point has the largest slope.

Explanation

Let the point be (x=a)

Let the slope function be m(x) .

y=1+40x33x5

Differentiate y with respect to x,

dydx=403x235x4=120x215x4

Hence the slope function is,

m(x)=120x215x4

The slope at (x=a) is,

m(a)=120a215a4

Differentiate m(a) with respect to a,

m(a)=1202a154a3=240a60a3

To find the critical values, m(a)=0 .

240a60a3=060a(a24)=060a(a2)(a+2)=0

Case 1:

m(a)<060a(a2)(a+2)<0

a>2 , a<0 and a>2 .

That is, 2<a<0 and a>2 .

Thus, m is increasing on (,2) and on (0,2)

Case 2:

m(a)>060a(a2)(a+2)>0

a<2 , a>0 and a<2

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