Proving the Existence of a Positive Real Number x Such That x3 2 Using the Intermediate Value Theorem

Let's consider the problem of proving the existence of a positive real number x such that x3 2. One effective method to solve this problem is to use the properties of continuous functions and the Intermediate Value Theorem. Below, we outline a step-by-step explanation of how to solve this problem.

Step 1: Define the Function and Equation

To prove the existence of a positive real number x such that x3 2, we define the function f(x) x3 - 2. The goal is to show that there exists a solution to the equation f(x) 0.

Step 2: Evaluate the Function at Specific Points

Next, we evaluate the function f(x) at specific points to establish the sign changes that will help us apply the Intermediate Value Theorem.

Calculate f(1):

f(1) 13 - 2 1 - 2 -1

Calculate f(2):

f(2) 23 - 2 8 - 2 6

Step 3: Apply the Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on different signs at the endpoints, then there exists at least one c in the interval [a, b] such that f(c) 0.

In our case, since f(1) -1 0 and f(2) 6 0, we can see that f (1) 0 and f(2) 0. By the Intermediate Value Theorem, there must be at least one point x in the interval [1, 2] where f(x) 0.

Step 4: Verify the Continuity of the Function

The function f(x) x3 - 2 is a polynomial, and all polynomials are continuous everywhere on the real line. Therefore, the function f(x) is continuous on the interval [1, 2].

Conclusion

Since f(x) is continuous on the interval [1, 2] and changes sign between f(1) 0 and f(2) 0, by the Intermediate Value Theorem, there exists at least one positive real number x in the interval [1, 2] such that f(x) 0. This means there exists a positive real number x such that x3 2.

This proof demonstrates the power and utility of the Intermediate Value Theorem in proving the existence of specific points in real numbers. It highlights the importance of the properties of continuous functions and the intermediate value property in solving such mathematical problems.