   Chapter 3.7, Problem 6E

Chapter
Section
Textbook Problem

# What is the minimum vertical distance between the parabolas y = x 2 + 1 and y = x − x 2 ?

To determine

To find:

The minimum vertical distance between the parabolas y=x2+1  and y=x-x2

Explanation

1) Concept:

The minimum vertical distance can be found by applying the first derivative test and critical numbers.

First derivative test for absolute extreme values - suppose that c is a critical number of a continuous function f defined on an interval.

a) If f'x>0 for all x<c and f'x<0 for all x>c, then fc is the absolute maximum value of f

b) If f'x<0 for all x<c and f'x>0 for all x>c, then fc is the absolute Minimum value of f

Critical number: 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.

2) Given:

y=x2+1 and y=x-x2

3) Calculation:

Put y=x2+1 in the equation y=x-x2

So, x2+1 =x-x2

2x2-x+1

Let Dx=</

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