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Chapter 3: Q. 31 (page 288)

In Exercises 31鈥�34, find the point on the graph of the function f that is closest to the point (a, b) by minimizing the square of the distance from the graph to the point.

f(x)=3x+1and the point (-2,1)

Short Answer

Expert verified

The point on the graph of the function that is close to the point is (-15,25).

Step by step solution

01

Step 1. Given Information.

The function:

f(x)=3x+1

The point:

(-2,1)

02

Step 2. Write the distance between thee function and the point.

The distance between (-2,1)and (x,y) is:
D(x)=(1-y)2+(-2-x)2=(1-3x-1)2+(-2-x)2=10x2+4x+4

03

Step 3. Find the derivative of the function.

D(x)=10x2+4x+4D'(x)=12(10x2+4x+4)-12.(20x+4)=2(10x+2)2(10x2+4x+4)=10x+210x2+4x+4

04

Step 4. Find the point close to the function and the point.

To find the point that is closest to the function and the point,

D'(x)=010x+210x2+4x+4=010x+2=010x=-2x=-210=-15

05

Step 5. Find y.

Substitute the value of x in the function to get y,

y=3x+1=3(-15)+1=-3+55=25

So the closest point is (-15,25)

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