Differentiability, Primitive, And Integrals Exercise 1 We Ha

Differentiability Primitive Integralsexercice 1we Have F A B

We are given a continuous function \(f : [a, b] \to \mathbb{R}\), which is differentiable on the open interval \((a, b)\). The function satisfies the conditions \(f(a) 0\), and adheres to the hypothesis (H): at any point \(x_0 \in (a, b)\) where \(f(x_0) = 0\), it also holds that \(f'(x_0) > 0\). The primary goal is to demonstrate that \(f\) has a unique zero \(\alpha \in (a, b)\).

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To prove the uniqueness of a zero of the function \(f\) under the given conditions, we begin by establishing that at least one such zero exists within the interval \((a, b)\). Since \(f\) is continuous on \([a, b]\), with \(f(a) 0\), the Intermediate Value Theorem guarantees that there exists some \(\alpha \in (a, b)\) such that \(f(\alpha) = 0\). This confirms the existence of at least one zero but does not ascertain its uniqueness.

Suppose, for contradiction, that there are two zeros \(\alpha\) and \(\beta\) in \((a, b)\), with \(a 0\). The existence of such a point follows from the fact that \(f\) changes sign within \((a, b)\), and by the properties of continuous functions, there must be a point \(\gamma\) between the zeros where \(f\) attains positive values.

Next, we construct two sequences \((a_n)\) and \((b_n)\) that are adjacent and converge to the common boundary point \(\omega\) where \(f(\omega) = 0\). We initialize with \(a_0 = \alpha\) and \(b_0 = \beta\), and recursively define the sequences via a bisection method:

  • If \(f(c) > 0\), where \(c = (a_n + b_n)/2\), then we set \(a_{n+1} = c\) and \(b_{n+1} = b_n\).
  • Otherwise, if \(f(c) \leq 0\), then \(a_{n+1} = a_n\) and \(b_{n+1} = c\).

This process ensures that the sequences \((a_n)\) and \((b_n)\) converge to the same point \(\omega\), with \(a_n \leq \omega \leq b_n\), and the intervals \([a_n, b_n]\) shrink to the point \(\omega\).

At the limit, since \(f\) is continuous, \(f(\omega) = 0\). To analyze the behavior of \(f\) near \(\omega\), we utilize a first-order Taylor expansion around \(\omega\), or a linear approximation. The difference quotient \(\frac{f(b_n) - f(a_n)}{b_n - a_n}\) converges to a limit as \(n \to \infty\). Calculus tells us that this limit equals \(f'(\omega)\), provided \(f\) is differentiable at \(\omega\).

From the construction, because \(f(a_n) 0\), the slope \(\frac{f(b_n) - f(a_n)}{b_n - a_n}\) tends to be non-positive when approaching the common zero, which implies that \(f'(\omega) \leq 0\). However, this contradicts hypothesis (H), which states that at any zero where \(f\) vanishes, it must be that \(f'(\omega) > 0\). Thus, our assumption of multiple zeros is false, and \(f\) has a unique \(\alpha \in (a, b)\) where \(f(\alpha) = 0\).

Application to Differential Equation and Further Analysis

Considering the additional application, suppose there exists a function \(y \in C^1(\mathbb{R}^+)\) such that \(y(0) = -10\) and it satisfies the differential equation:

\[

y'(t) = \ln(1 + t^4 + t) - \sin \left( y(t)^3 \right).

\]

To analyze the existence and uniqueness of a zero of \(y\), observe that \(y\) is continuously differentiable, ensuring that the behavior of \(y\) can be studied via standard differential equation techniques. First, note that for sufficiently large \(t\), since \(\ln(1 + t^4 + t)\) dominates the sinusoidal term, \(y'\) tends to be positive, implying that \(y(t)\) eventually increases without bound. Conversely, near \(t=0\), \(y(0) = -10\), and the initial derivative dictates the trend of \(y\).

To show that \(y\) has a unique zero on \(\mathbb{R}^+\), one can argue as follows: If the derivative \(y'(t) \geq 1\) for all \(t \geq T\), then \(y(t)\) will increase at least linearly, crossing zero exactly once. Analyzing whether such a \(T\) exists involves examining the bounds of \(\ln(1 + t^4 + t)\) and \(\sin(y(t)^3)\). Since \(\sin\) is bounded between \(-1\) and 1, eventual dominance by the logarithm guarantees \(y'\) becomes positive beyond some \(T\). This guarantees at least one zero (by the Intermediate Value Theorem) and only one zero due to the strict monotonicity of \(y\) beyond that point.

Overall, the combination of the properties of \(f\) and \(y\) demonstrates the powerful application of differentiability and the Mean Value Theorem in analyzing zeros of functions, confirming their uniqueness under appropriate hypotheses.

References

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  • Coddington, E. A., & Levinson, N. (1955). Theory of Ordinary Differential Equations. McGraw-Hill.
  • Courant, R., & John, F. (1989). Introduction to Calculus and Analysis. Springer.
  • Walter, W. (1998). Ordinary Differential Equations. Springer.
  • Strang, G. (2016). Calculus: Early Transcendentals. Wellesley-Cambridge Press.
  • Perko, L. (2001). Differential Equations and Dynamical Systems. Springer.
  • Leithold, L. (2012). Calculus: Concepts and Contexts. Pearson Education.
  • Goldstein, H. (1980). Classical Mechanics. Addison-Wesley.
  • Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications. Wiley-Interscience.
  • Blanchard, P., Devaney, R. L., & Kukka, D. (2012). Differential Equations. Brooks Cole.

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