Given as true f(x) = (1+1/x)^x is strictly increasing for x>=1

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In summary, we are given the function f(x) = (1+1/x)^x, which is strictly increasing for x>=1 and has a horizontal asymptote at y=e. We are asked to prove that (n/3)^n < n! < (n/2)^n for all integers n>=6. To do this, we can use the function f(x) as a substitute for n. The base case can be chosen as n=6, and then we can use induction to show that the inequality holds for all subsequent values of n.
  • #1
mebigp
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Homework Statement


Given as true f(x) = (1+1/x)^x is strictly increasing for x>=1 and that f(x) has horizontal asymptote y=e.

Prove that (n/3)^n< n! <(n/2)^n for all integers n>=6 ?

Homework Equations


The Attempt at a Solution


f(x)=(1+1/x)^x is increasing and approach e
prove (n/3)^n< n! <(n/2)^n for all n>=6

So I attempt to use f(x) to replace n but that will not work for the base case 1 because n>6
 
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  • #2


mebigp said:

Homework Statement


Given as true f(x) = (1+1/x)^x is strictly increasing for x>=1 and that f(x) has horizontal asymptote y=e.

Prove that (n/3)^n< n! <(n/2)^n for all integers n>=6 ?

Homework Equations





The Attempt at a Solution


f(x)=(1+1/x)^x is increasing and approach e
prove (n/3)^n< n! <(n/2)^n for all n>=6

So I attempt to use f(x) to replace n but that will not work for the base case 1 because n>6
The base case doesn't have to be n = 1. Use n = 6 for your base case.
 
  • #3


Thanks Mark44

So I use k=F(x) from first function

Base k=6;

(k/3)^k< k! <(k/2)^k

64<180<729 true

K+1->;
((k+1)/3)^(k+1)< (k+1)! <((k+1)/2)^(k+1)

But why was I given as true f(x) = (1+1/x)^x all I did was make K=f(x)
Do I have to work out how to get a 6 from that function.( I don't think its possible)
 
Last edited:

FAQ: Given as true f(x) = (1+1/x)^x is strictly increasing for x>=1

1. What does it mean for a function to be strictly increasing?

A function is strictly increasing if the values of the function increase as the input values increase. In other words, as the x-values increase, the corresponding y-values also increase.

2. How do we know that f(x) = (1+1/x)^x is strictly increasing?

We can prove that f(x) = (1+1/x)^x is strictly increasing for x>=1 by taking the derivative of the function and showing that it is always positive for x>=1. This means that the function is always increasing and never decreasing for all values of x greater than or equal to 1.

3. Why is the restriction x>=1 necessary for f(x) = (1+1/x)^x to be strictly increasing?

The restriction x>=1 is necessary because the function (1+1/x)^x is only strictly increasing for values of x greater than or equal to 1. For values of x less than 1, the function is not strictly increasing and can even be decreasing at certain points.

4. Can you provide a graphical representation of f(x) = (1+1/x)^x to illustrate its strict increasing nature?

Yes, the graph of f(x) = (1+1/x)^x will show a steadily increasing curve for all values of x greater than or equal to 1. This means that as x increases, the corresponding y-values on the graph will also increase, demonstrating the strict increasing nature of the function.

5. Are there any other ways to prove that f(x) = (1+1/x)^x is strictly increasing for x>=1?

Yes, there are other ways to prove this statement. One way is to use the mean value theorem, which states that if a function has a positive derivative for all values in an interval, then the function must be strictly increasing on that interval. Another way is to use the definition of a strictly increasing function and show that it holds true for f(x) = (1+1/x)^x for all x-values greater than or equal to 1.

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