Don't undertstand what they are asking

[Chas3down]

Homework Statement

The goal of this problem is to use the Intermediate Value Theorem to prove that there exists a positive number c which is the fifth root of 2.

Another way of expressing this is that we would like to find a positive root \,c of the continuous function f(x) = x^5 - ? .
Note: Fill in the box with an appropriate constant to complete the definition of the function f(x).

Homework Equations

The Attempt at a Solution

I tried 32, 1/32, which I thought they were just asking for 2^5? But I guess not.. I understand what the intermediate value theorem is, but not in the context of what they are asking.

 

[pasmith]

Homework Statement

The goal of this problem is to use the Intermediate Value Theorem to prove that there exists a positive number c which is the fifth root of 2.

Another way of expressing this is that we would like to find a positive root \,c of the continuous function f(x) = x^5 - ? .
Note: Fill in the box with an appropriate constant to complete the definition of the function f(x).

Homework Equations

The Attempt at a Solution

I tried 32, 1/32, which I thought they were just asking for 2^5? But I guess not.. I understand what the intermediate value theorem is, but not in the context of what they are asking.

You need to have f(x) = 0 when x = 2^{1/5}, not when x = 2^5.

 

[johnqwertyful]

The root of x^5-C is when x^5-C=0. You want to find some constant C such that the root is the fifth root of 2.

 

[Chas3down]

Oh got it, 2^1/5^5 = 2 so c = 2 thanks!

 

[johnqwertyful]

Oh got it, 2^1/5^5 = 2 so c = 2 thanks!

C=2, but I don't get your logic.

 

[Ray Vickson]

Oh got it, 2^1/5^5 = 2 so c = 2 thanks!

No, you are not done! You are assuming the existence of 2^(1/5), but you cannot do that: the question is asking you to prove that such a 5th root actually exists.

 

[A David]

^ Listen to Ray. You haven't used the theorem! You may want to think about, say, f(0) and f(2), and how we can use these values and the theorem to show that 2 must have a positive fifth root.

 

[HallsofIvy]

C=2, but I don't get your logic.

He wrote "2^1/5^5" but meant (2^1/5)^5.