Show that f(x) = x^3 - 2x - 1 has a real zero between 1 and 2.

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Step 1: Understand the problem. We need to show that the function \( f(x) = x^3 - 2x - 1 \) has a real zero between \( x = 1 \) and \( x = 2 \). This means we need to find values of \( f(x) \) at these points and check for a sign change.

Step 2: Calculate \( f(1) \). Substitute \( x = 1 \) into the function: \( f(1) = 1^3 - 2(1) - 1 \). Simplify this expression to find the value of \( f(1) \).

Step 3: Calculate \( f(2) \). Substitute \( x = 2 \) into the function: \( f(2) = 2^3 - 2(2) - 1 \). Simplify this expression to find the value of \( f(2) \).

Step 4: Analyze the results. Check the signs of \( f(1) \) and \( f(2) \). If \( f(1) \) and \( f(2) \) have opposite signs, then by the Intermediate Value Theorem, there is at least one real zero between \( x = 1 \) and \( x = 2 \).

Step 5: Conclude. If a sign change is observed, conclude that \( f(x) = x^3 - 2x - 1 \) has a real zero between \( x = 1 \) and \( x = 2 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function takes on two values at two points, then it must take on every value between those two points at least once. This theorem is essential for proving the existence of a real zero in the given interval, as it guarantees that if the function changes signs between two points, a root exists in that interval.

Continuous Functions

A continuous function is one where small changes in the input result in small changes in the output, meaning there are no breaks, jumps, or holes in the graph. The function f(x) = x^3 - 2x - 1 is a polynomial, and all polynomial functions are continuous, which is a critical property for applying the Intermediate Value Theorem.

Evaluating Function Values

To apply the Intermediate Value Theorem, we need to evaluate the function at the endpoints of the interval. By calculating f(1) and f(2), we can determine if the function changes sign between these two points, which indicates the presence of a real zero. This step is crucial for establishing the conditions required by the theorem.