# How to solve this IVT problem??

• Oct 18th 2009, 01:33 PM
skboss
How to solve this IVT problem??
How do I solve this problem?

Use IVT (Intermediate Value Theorem) to prove the following statement:

2^x = bx has a solution if b>2

Attempt: I know that IVT says that a continuous function has values in every single points, but if I don't the value in front of x how am I supposed to solve that?
• Oct 18th 2009, 01:46 PM
HallsofIvy
Quote:

Originally Posted by skboss
How do I solve this problem?

Use IVT (Intermediate Value Theorem) to prove the following statement:

2^x = bx has a solution if b>2

Attempt: I know that IVT says that a continuous function has values in every single points, but if I don't the value in front of x how am I supposed to solve that?

Let f(x)= \$\displaystyle 2^x- bx\$. Then f(1)= 2- b< 0. Whatever b is, there exist n such that \$\displaystyle 2^n> b\$. For that n, \$\displaystyle f(2^n)= 2^{2^n}- b(2^n)= 4^n- 2^nb= 2^n(2^n- b)> 0\$.

Now use the IVT
• Oct 18th 2009, 02:03 PM
skboss
Quote:

Originally Posted by HallsofIvy
Let f(x)= \$\displaystyle 2^x- bx\$. Then f(1)= 2- b< 0. Whatever b is, there exist n such that \$\displaystyle 2^n> b\$. For that n, \$\displaystyle f(2^n)= 2^{2^n}- b(2^n)= 4^n- 2^nb= 2^n(2^n- b)> 0\$.

Now use the IVT

Thank you very much, but I got lost after the 2nd step when u introduce n. If you can explain that, that would be great. Thanks again.
• Oct 19th 2009, 04:19 AM
HallsofIvy
Why part do you need explained?

Do you understand why I wanted to be able to say that f(1)< 0 and that f(some number) > 0?

Do you understand why \$\displaystyle 2^{2^n}= 4^n= (2^n)(2^n)\$?
The text passage contains a mathematical question and an answer provided by another user. The question is about using the Intermediate Value Theorem to prove that the equation \(2^x = bx\) has a solution if \(b > 2\). The answer provides a detailed explanation of how to apply the IVT to solve the problem.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# How to solve this IVT problem??

• Oct 18th 2009, 01:33 PM
skboss
How to solve this IVT problem??
How do I solve this problem?

Use IVT (Intermediate Value Theorem) to prove the following statement:

2^x = bx has a solution if b>2

Attempt: I know that IVT says that a continuous function has values in every single points, but if I don't the value in front of x how am I supposed to solve that?
• Oct 18th 2009, 01:46 PM
HallsofIvy
Quote:

Originally Posted by skboss
How do I solve this problem?

Use IVT (Intermediate Value Theorem) to prove the following statement:

2^x = bx has a solution if b>2

Attempt: I know that IVT says that a continuous function has values in every single points, but if I don't the value in front of x how am I supposed to solve that?

Let f(x)= \$\displaystyle 2^x- bx\$. Then f(1)= 2- b< 0. Whatever b is, there exist n such that \$\displaystyle 2^n> b\$. For that n, \$\displaystyle f(2^n)= 2^{2^n}- b(2^n)= 4^n- 2^nb= 2^n(2^n- b)> 0\$.

Now use the IVT
• Oct 18th 2009, 02:03 PM
skboss
Quote:

Originally Posted by HallsofIvy
Let f(x)= \$\displaystyle 2^x- bx\$. Then f(1)= 2- b< 0. Whatever b is, there exist n such that \$\displaystyle 2^n> b\$. For that n, \$\displaystyle f(2^n)= 2^{2^n}- b(2^n)= 4^n- 2^nb= 2^n(2^n- b)> 0\$.

Now use the IVT

Thank you very much, but I got lost after the 2nd step when u introduce n. If you can explain that, that would be great. Thanks again.
• Oct 19th 2009, 04:19 AM
HallsofIvy
Why part do you need explained?

Do you understand why I wanted to be able to say that f(1)< 0 and that f(some number) > 0?

Do you understand why \$\displaystyle 2^{2^n}= 4^n= (2^n)(2^n)\$?
user
The text passage contains a mathematical question and an answer provided by another user. The question is about using the Intermediate Value Theorem to prove that the equation \(2^x = bx\) has a solution if \(b > 2\). The answer provides a detailed explanation of how to apply the IVT to solve the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER