Show That The Equation Has Exactly One Real Root

Show that the equation has exactly 1 real root

Although the user provided a collection of multiple prompts and partial instructions, the core assignment question to address is: "Show that the equation has exactly 1 real root." This concise instruction points to a mathematical proof or analysis demonstrating the uniqueness of a real solution for a given equation.

Since the specific equation is not explicitly provided in the instructions, we will assume it is a continuous function \(f(x)\) defined over the real numbers, and our task is to establish that this function has exactly one real root. Typical methods for such proofs include demonstrating that the function crosses the x-axis only once, by analyzing its derivative, monotonicity, and limits.

Paper For Above instruction

To rigorously demonstrate that a given equation has exactly one real root, it is essential to analyze the properties of the function involved—namely continuity, differentiability, and the behavior at the limits. The general approach involves several steps: confirming the existence of at least one root, and then establishing its uniqueness.

Identification of the function

Suppose the function \(f(x)\) is continuous on \(\mathbb{R}\) and differentiable on \(\mathbb{R}\). To establish the existence of at least one root, the Intermediate Value Theorem (IVT) can be employed if \(f\) takes on values of opposite signs at two points. For example, if \(f(a) 0\) for some \(a

To prove that this root is unique, one can analyze the derivative \(f'(x)\). If \(f'(x)\) maintains a consistent sign (either always positive or always negative), then \(f(x)\) is monotonic (strictly increasing or decreasing) across its domain. Monotonic functions intersect the x-axis at most once, ensuring the uniqueness of the root.

Thus, the critical components of the proof include:

1. Showing the function crosses the x-axis (root existence).
2. Proving the derivative does not change sign (strict monotonicity), which guarantees exactly one crossing point (root uniqueness).

Application of the proof steps

Assuming \(f(x)\) is such that \(\lim_{x \to -\infty}f(x) = -\infty\) and \(\lim_{x \to +\infty}f(x) = +\infty\), the IVT assures at least one root. Next, analyze \(f'(x)\). If \(f'(x) > 0\) or \(f'(x)

For instance, consider the specific case \(f(x) = x^3 - x + 1\). It's continuous and differentiable everywhere. Its derivative \(f'(x) = 3x^2 - 1\), which is never negative for \(x \geq \frac{1}{\sqrt{3}}\) and positive elsewhere. By analyzing \(f'(x)\), one can conclude whether the function is monotonic over possible ranges.

In conclusion, demonstrating that the derivative does not change sign (or changes sign only once) provides a solid basis for asserting the function has exactly one real root. The argument hinges on the function's monotonicity, limits, and the Intermediate Value Theorem.

References

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