Proving Unique Solution for f(x) = 0 on (a,b) with f'(x) ≠ 0

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In summary, assuming that f is continuous and differentiable on [a,b], with f′(x) ≠ 0 on (a,b) and f(a) and f(b) having different signs, the equation f(x) = 0 has a unique solution in (a,b). This can be proven by first showing that there is a solution in (a,b), and then assuming the contrary to prove that there is only one such solution.
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Assume that f is continuous on [a,b] and differentiable on (a,b). Assume also that f′(x) ≠ 0 on (a,b) and f(a) and f(b) have different signs. Show that the equation f (x) = 0 has a unique solution in (a, b).


I'm not really sure how to even start this proof. Do I need to use the Intermediate Value Theorem? Any help would be great!
 
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msell2 said:
Assume that f is continuous on [a,b] and differentiable on (a,b). Assume also that f′(x) ≠ 0 on (a,b) and f(a) and f(b) have different signs. Show that the equation f (x) = 0 has a unique solution in (a, b).


I'm not really sure how to even start this proof. Do I need to use the Intermediate Value Theorem? Any help would be great!

You actually need to prove two things: (1) there is a solution x in (a,b); and (2) there is only one such solution.

Do you know how to get (1)?

For (2): assume the contrary and see what happens.
 

FAQ: Proving Unique Solution for f(x) = 0 on (a,b) with f'(x) ≠ 0

What is a unique solution proof?

A unique solution proof is a mathematical method used to show that a given equation or problem has only one possible solution. It involves demonstrating that all other potential solutions either do not exist or do not satisfy the conditions of the problem.

Why is it important to prove a unique solution?

Proving a unique solution is important because it ensures that there is only one correct answer to a problem, eliminating any ambiguity or uncertainty. This is particularly important in fields such as science and engineering, where accurate and reliable solutions are crucial.

What are the steps involved in a unique solution proof?

The first step is to clearly define the problem and any constraints or conditions that must be met. Next, the proof typically involves using logical reasoning and mathematical techniques to show that any other potential solutions do not meet the specified criteria. Finally, the proof must be presented in a clear and rigorous manner.

What types of problems can be solved using unique solution proofs?

Unique solution proofs can be used for a wide range of problems, including algebraic equations, geometric constructions, optimization problems, and more. Essentially, any problem that requires a single, definitive solution can be approached using unique solution proofs.

Are there any limitations to unique solution proofs?

While unique solution proofs can be a powerful tool for solving problems, they are not appropriate for every situation. In some cases, a problem may have multiple valid solutions, making it impossible to prove a unique solution. Additionally, unique solution proofs may not be the most efficient or practical method for solving certain types of problems.

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