   Chapter 3.7, Problem 57E

Chapter
Section
Textbook Problem

# What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve y = 3 / x at some point?

To determine

To find:

The shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve y=3x

Explanation

1) Concept:

Equation of the tangent line at the point (x1, y1) is given by

y2-y1=m(x2-x1)

First derivative test- if f' changes from negative to positive at c then f  has local minimum at c

Second derivative test- Suppose f'' is continuous near c

i. If f'c=0 and f''c>0 then f  has local minimum at c

ii. If f'c=0 and f''c<0 then f  has local maximum at c

2) Given:

The curve y=3x

3) Calculation:

Given that

y=3x

Let (x, 3x) be the point of the curve y=3x

Differentiate the above equation using the formula

y'=-3x2

Equation of the tangent line at the point (x, 3x) is given by

y2-3x=-3x2(x2-x)

By setting x2=0 and y2=0 we will get endpoints

Put x2=0 in y2-3x=-3x2(x2-x)

y2-3x=-3x2(0-x)

Simplify

y2-3x=3x

y2-3x+3x=3x+3x

Simplify

y2=6x

This is y- intercept

And Put y2=0 in y2-3x=-3x2(x2-x)

0-3x=-3x2(x2-x)

Simplify

x2=2x

This is x- intercept

The distance D of the line segment that has endpoints at the intercepts is

D=2x-02+0-6x2

Simplify

D=2x2+6x</

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