# Prove Existence Unique Real Solution

• knowLittle
In summary, the "Prove Existence Unique Real Solution" problem is a mathematical puzzle that asks whether an equation has a unique real solution. To prove the existence of a unique real solution, one must show that there is at least one real solution and that there cannot be more than one. This is important in mathematics and physics for making predictions and ensuring accuracy. There can only be one unique real solution to an equation, and common techniques used to prove this include the Intermediate Value Theorem, Rolle's Theorem, and the Mean Value Theorem.
knowLittle

## Homework Statement

Prove Existence Unique Real Solution to
## x^{3} + x^{2} -1 =0 ## between ## x= \frac{2}{3} \text{and} x=1##

## The Attempt at a Solution

## x^{2} ( x+1) =1 ##
I know that the solution is x =0.75488, but this came from some website. How do I find this number without a calculator?

You don't actually need to find the solution. You just need to know it exists. Think intermediate value theorem.

1 person
The intermediate value theorem shows that there exist a solution betweem 2/3 and 1. The mean value theorem shows that there is only one.

## 1. What is the "Prove Existence Unique Real Solution" problem?

The "Prove Existence Unique Real Solution" problem is a mathematical puzzle that asks whether a given equation has a unique real solution, meaning there is only one value of the variable that satisfies the equation. This problem is commonly encountered in calculus and other advanced mathematical fields.

## 2. How do you prove the existence of a unique real solution?

To prove the existence of a unique real solution, one must show that the equation has at least one real solution and that there cannot be more than one. This can be done using various mathematical techniques such as the Intermediate Value Theorem, Rolle's Theorem, and the Mean Value Theorem.

## 3. What is the importance of proving the existence of a unique real solution?

Proving the existence of a unique real solution is important in many areas of mathematics and physics. It allows us to determine the validity of certain equations and make predictions about the behavior of systems. It also helps to ensure the accuracy of numerical calculations and mathematical models.

## 4. Can there be more than one unique real solution to an equation?

No, there can only be one unique real solution to an equation. This means that there is only one value of the variable that makes the equation true. If there were more than one solution, the equation would not be considered to have a unique real solution.

## 5. What are some common techniques used to prove the existence of a unique real solution?

As mentioned earlier, some common techniques used to prove the existence of a unique real solution include the Intermediate Value Theorem, Rolle's Theorem, and the Mean Value Theorem. Other techniques include using algebraic manipulation and graphical analysis to show that there is only one point of intersection between two equations.

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