   Chapter 3.7, Problem 5E

Chapter
Section
Textbook Problem

# What is the maximum vertical distance between the line y = x + 2 and the parabola y = x 2 for − 1 ≤ x ≤ 2 ?

To determine

To find:

The maximum vertical distance between the line y=x+2 and the parabola y=x2 for -1x2

Explanation

1) Concept:

i. A critical number of a function f   is a number c in the domain of f  Such that either  f'c=0 or f'c does not exist.

ii. Concavity test:

If f"(x)>0 then the graph of f is concave upward

If f"(x)<0 then the graph of f is concave downward

2) Given:

y=x+2 and y=x2; -1 x 2

3) Calculation:

Put y=x+2  in the equation y=x2

So, x2=x+2

Subtract x+2  from both sides

x2-x+2=x+2-x+2x2-x-2

Let fx=x2-x-2

Differentiate fx

f'x=2x-1

To find critical point equate f'x=0 f'x=2x-

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