Intermediate Value Theorem proof

In summary, the conversation discusses the proof of existence and uniqueness of nonnegative nth roots for any nonnegative real number. The hint given is to use the intermediate value theorem to show the contradiction of assuming two nonnegative real numbers have the same root. This approach proves the statement.
  • #1
Anisotropic Galaxy
19
0
Hi, can someone please help me here?

Prove that every nonnegative real number x has a unique nonnegative nth root x^(1/n).

The problem gives a hint - existence of x^(1/n) can be seen by applying intermediate value theorem to function f(t) = t^n for t>= 0.

But I still don't get it - proofs are such a hurdle for me. I'm hoping that I can get over that hurdle...soon...

Thanks!
 
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  • #2
Anisotropic Galaxy said:
Hi, can someone please help me here?
Prove that every nonnegative real number x has a unique nonnegative nth root x^(1/n).
The problem gives a hint - existence of x^(1/n) can be seen by applying intermediate value theorem to function f(t) = t^n for t>= 0.
But I still don't get it - proofs are such a hurdle for me. I'm hoping that I can get over that hurdle...soon...
Thanks!

try to get a contradiction of the statement you want to prove. so, assume that there exists two non negative real which have the same root. then you should be able to use the IVT to show that the two real numbers are actually the same number and that is a contradiction of your assumption which proves the proof statement.
 

Related to Intermediate Value Theorem proof

1. What is the Intermediate Value Theorem?

The Intermediate Value Theorem is a fundamental theorem in calculus that states that if a continuous function takes on two values at two points, then it must also take on all values between those two points. In other words, if f(a) and f(b) are two values of a continuous function f(x), then f(x) must take on all values between f(a) and f(b) for some value of x between a and b.

2. What is the importance of the Intermediate Value Theorem?

The Intermediate Value Theorem is important because it guarantees the existence of roots or solutions to equations. It is also used in many proofs and is a fundamental concept in calculus and real analysis.

3. How is the Intermediate Value Theorem proved?

The Intermediate Value Theorem is typically proved using a proof by contradiction. This involves assuming that the theorem is false and then showing that this leads to a contradiction. This contradiction then proves that the theorem must be true.

4. Can the Intermediate Value Theorem be applied to all functions?

No, the Intermediate Value Theorem can only be applied to continuous functions. A continuous function is one that does not have any breaks or jumps in its graph. Moreover, both the function and its input must be real numbers.

5. How is the Intermediate Value Theorem used in real-world applications?

The Intermediate Value Theorem is used in many real-world applications, such as physics, engineering, and economics. It is used to prove the existence of solutions to equations and to analyze the behavior of functions. For example, it can be used to determine the existence of a solution to a physical problem, such as finding the position of an object at a specific time, or to analyze the behavior of supply and demand in economics.

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