1.Suppose that f : [0, 1] −→ [0, 1] is a continuous function. Prove that f has a fixed point in [0, 1], i.e., there is at least one real number x ∈ [0, 1] such that f(x) = x.

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....
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1.Suppose that f : [0, 1] −→ [0, 1] is a continuous function. Prove that f has a
fixed point in [0, 1], i.e., there is at least one real number x ∈ [0, 1] such that
f(x) = x.

 

2.The axes of two right circular cylinders of radius a intersect at a right angle.
Find the volume of the solid of intersection of the cylinders.

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