A continuous function ƒ : [0, 1] → R satisfies ƒ(0) f(1). Show that for each integer n ≥ 1 there exits x such that f(x+(1/n)) = f(x). Is the same statement true for numbers other than 1/n? =
A continuous function ƒ : [0, 1] → R satisfies ƒ(0) f(1). Show that for each integer n ≥ 1 there exits x such that f(x+(1/n)) = f(x). Is the same statement true for numbers other than 1/n? =
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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Step 1: Proof
Let fix
Consider the continuous function and suppose that on
Then by intermediate value theorem, does not changes sign in .
Without losing of generality if , then adding the inequalities
yields which is a contradiction of our hypothesis that is
Thus there exist a point satisfying
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