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Chapter 7: Problem 8

(Wronskians) The Wronskian of two differentiable functions \(f\) and \(g\) is the determinant $$ W(f, g)=\left|\begin{array}{cc} f(x) & g(x) \\ f^{\prime}(x) & g^{\prime}(x) \end{array}\right| $$ Prove that if \(W(f, g)\) does not vanish on an interval \(I\) and \(f\left(x_{1}\right)=f\left(x_{2}\right)=\) 0 for points \(x_{1}

Short Answer

Expert verified
There exists a point \(x_3\) between \(x_1\) and \(x_2\) where \(g(x_3) = 0\).

Step by step solution

01

Define the Wronskian

The Wronskian for two functions \(f\) and \(g\) is given by the determinant \[W(f, g) = \begin{vmatrix} f(x) & g(x) \ f'(x) & g'(x) \end{vmatrix} = f(x)g'(x) - f'(x)g(x).\] We need to show that if this Wronskian \( W(f, g) \) is non-zero for an interval \( I \), and \( f(x) \) has two zeros at points \( x_1 \) and \( x_2 \), then \( g(x) \) must vanish at some point between them.
02

Use Rolle's Theorem

Since \( f(x_1) = f(x_2) = 0 \), by Rolle's Theorem, there exists at least one point \( c \) in \((x_1, x_2)\) where \( f'(c) = 0 \).
03

Analyze the Wronskian at Point c

Substitute \( x = c \) into the Wronskian formula: \[W(f, g) = f(c)g'(c) - f'(c)g(c).\] Since \( f'(c) = 0 \) from Step 2, and \( f(c) = 0 \) (because \(c\) is a point on the function path between two zeros and Rolle's Theorem doesn't affect \(g(x)\)), the Wronskian simplifies to \((-0) = 0\), which contradicts our non-zero assumption.
04

Conclude Existence of g(x3) = 0

Therefore, for the Wronskian to remain non-zero, it must be that \( g(c) = 0 \). Thus, there exists a point \( x_3 = c \) in the interval \( (x_1, x_2) \) such that \( g(x_3) = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differentiable Functions
Differentiable functions are fundamental to calculus and mathematical analysis. A function is said to be differentiable at a point if it has a derivative there. In simpler terms, this means that the function has a defined slope, or rate of change, at that point. Differentiation can be thought of as finding the instant rate of change of the function as you zoom in ever closer to any specific point on its curve.
For a function to be differentiable, it must be smooth and continuous around the vicinity of the point. This means there shouldn鈥檛 be any sharp corners or jumps, which makes differentiation analogous to the ability of drawing a tangent line without any interruptions.
When dealing with two differentiable functions, such as in the calculation of the Wronskian, the differentiability implies we can reliably find each function鈥檚 derivative and use them in various operations like building determinants.
Rolle's Theorem
Rolle's Theorem is a pivotal result in calculus that provides insights into the behavior of differentiable functions on a closed interval. It states that if a differentiable function has equal values at two distinct points in an interval, there must be at least one point in between where the derivative of the function is zero. In essence, the theorem assures the existence of a horizontal tangent line, or simply a point where the slope of the function is flat, between the selected points.
This concept aids in proving scenarios like the exercise involving the Wronskian by identifying points where functions momentarily don't change direction. In the context of the given solution, because the function turned zero at the interval endpoints, Rolle's Theorem was applied to demonstrate that its derivative also becomes zero at least once between these points. This characteristic was critical in further analyzing the continuity and changes in the Wronskian value.
Determinant
The determinant is a calculated value that can determine whether a system of linear equations has a unique solution, and is a key player in linear algebra. For a 2x2 matrix, the determinant is computed using the formula: \[ \text{det} \begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc \]It serves as an indicator of linear independence within a matrix. If the determinant is non-zero, the matrix is invertible, and the vectors involved are independent.
In the context of Wronskians, we use determinants to understand the linear relationship between differentiable functions. A non-zero Wronskian implies the functions are linearly independent on an interval. In the exercise鈥檚 solution, this property of the Wronskian is crucial for proving the zero-crossing of the second function between two zeros of the first. The Wronskian determinant's specific value helps establish whether functions have value intersection or not over a specified interval.

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Most popular questions from this chapter

Suppose that \(f\) is continuous on \([0,1]\), differentiable on \((0,1)\), and \(f(0)=0\) and \(f(1)=1\). For every integer \(n\) show that there must exist \(n\) distinct points \(\xi_{1}, \xi_{2}, \ldots, \xi_{n}\) in that interval so that $$ \sum_{k=1}^{n} \frac{1}{f^{\prime}\left(\xi_{k}\right)}=n $$

Suppose \(f\) satisfies the hypotheses of the mean value theorem on \([a, b]\). Let \(S\) be the set of all slopes of chords determined by pairs of points on the graph of \(f\) and let $$ D=\left\\{f^{\prime}(x): x \in(a, b)\right\\} $$ (a) Prove that \(S \subset D\). (b) Give an example to show that \(D\) can contain numbers not in \(S\).

Let \(f\) be continuous on an interval \([a, b]\) and twice differentiable on \((a, b)\) with a second derivative that never is zero. Show that \(f\) maps \([a, b]\) two-one onto some other interval; that is, there are at most two points in \([a, b]\) mapping into any one value in the range of \(f\).

Let \(f\) be convex on an open interval \((a, b)\). Show that then there are only two possibilities. Either (i) \(f\) is nonincreasing or nondecreasing on the entire interval \((a, b)\) or else (ii) there is a number \(c\) so that \(f\) is nonincreasing on \((a, c]\) and nondecreasing on \([c, b)\).

(Derived Numbers) The Dini derivates are sometimes called "extreme unilateral derived numbers." Let \(\lambda \in[-\infty, \infty] .\) Then \(\lambda\) is a derived number for \(f\) at \(x_{0}\) if there exists a sequence \(\left\\{x_{k}\right\\}\) with \(\lim _{k \rightarrow \infty} x_{k}=x_{0}\) such that $$ \lambda=\lim _{k \rightarrow \infty} \frac{f\left(x_{k}\right)-f\left(x_{0}\right)}{x_{k}-x_{0}} $$ (a) For the function \(f(x)=\left|x \cos x^{-1}\right|, f(0)=0\), show that every number in the interval \([-1,1]\) is a derived number for \(f\) at \(x=0 .\) Show that the two extreme derived numbers from the right are 0 and 1 , and the two from the left are \(-1\) and 0 . (b) Show that a function has a derivative at a point if and only if all derived numbers at that point coincide. (c) Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and let \(x_{0} \in \mathbb{R}\). Prove that if \(f\) is continuous on \(\mathbb{R}\), then the set of derived numbers of \(f\) at \(x_{0}\) consists of either one or two closed intervals (that might be degenerate or unbounded). Give examples to illustrate the various possibilities.
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