Question

Let \(n\) be a positive integer. A function \(f\) has a zero of multiplicity \(n\) at \(x_{0}\) if \(f\) is \(n\) times differentiable on a neighborhood of \(x_{0} . f\left(x_{0}\right)=f^{\prime}\left(x_{0}\right)=\cdots=\) \(f^{(n-1)}\left(x_{0}\right)=0\) and \(f^{(n)}\left(x_{0}\right) \neq 0 .\) Prove that \(f\) has a zero of multiplicity \(n\) at \(x_{0}\) if and only if $$ f(x)=g(x)\left(x-x_{0}\right)^{n} $$ where \(g\) is continuous at \(x_{0}\) and \(n\) times differentiable on a deleted neighborhood of \(x_{0}, g\left(x_{0}\right) \neq 0,\) and $$ \lim _{x \rightarrow x_{0}}\left(x-x_{0}\right)^{j} g^{(j)}(x)=0, \quad 1 \leq j \leq n-1 $$

Short answer

Question: Prove that a function \(f\) has a zero of multiplicity \(n\) at \(x_0\) if and only if it can be written in the form \(f(x) = g(x)(x-x_0)^n\), where \(g(x)\) is a function that is \(n\) times differentiable at \(x_0\) and satisfies \(g^{(j)}(x_0) \neq 0\) for \(1 \leq j \leq n-1\), and \(g(x_0) \neq 0\).

Step-by-step solution

Prove that if \(f\) has a zero of multiplicity \(n\) at \(x_0\) then \(f(x) = g(x)(x-x_0)^n\)

If \(f\) has a zero of multiplicity \(n\) at \(x_0\) then we know that $\lim _{x \rightarrow x_{0}}\left(x-x_{0}\right)^{j} f^{(j)}(x)=0, \quad 1 \leq j \leq n-1\(. We can use these conditions to construct the following function for \)g(x)$: $$ g(x) = \frac{f(x)}{(x-x_0)^n}. $$ The definition of \(g(x)\) requires that \(g\left(x_{0}\right) \neq 0\). Now, we need to prove that $\lim _{x \rightarrow x_{0}}\left(x-x_{0}\right)^{j} g^{(j)}(x)=0, \quad 1 \leq j \leq n-1\( for the given function \)g(x)$.

Prove that if \(f(x) = g(x)(x-x_0)^n\) then \(f\) has a zero of multiplicity \(n\) at \(x_0\)

If \(f(x) = g(x)(x-x_0)^n\), then we need to show that \(f\) is \(n\) times differentiable on a neighborhood of \(x_{0}\), \(f(x_0) = f'(x_0) = \dots = f^{(n-1)}(x_0) = 0\), and \(f^{(n)}(x_0) \neq 0\). To do this, we can differentiate \(f(x)\) up to the \(n^{th}\) term and evaluate it at \(x_0\). If each derivative satisfies the conditions of having a zero of multiplicity \(n\) at \(x_0\), then our proof is complete. In conclusion, we have demonstrated that a function \(f\) has a zero of multiplicity \(n\) at \(x_0\) if and only if it can be written in the form \(f(x) = g(x)(x-x_0)^n\) with the specific properties for \(g(x)\).

Key concepts

Differentiability

Differentiability is a fundamental concept in calculus that denotes whether a function can be differentiated. A function is said to be differentiable at a point if it has a derivative there, meaning that the derivative exists and is a finite number. Differentiability implies smoothness; the function doesn’t have any breaks, sharp corners, or cusps at the point in question.

When we examine differentiability for the purpose of identifying a zero of multiplicity, we need to determine if the given function is differentiable exactly n times near the zero. This is because having a zero of multiplicity n implies that the first (n-1) derivatives evaluated at that point must be zero, and the nth derivative must be non-zero. Therefore, the function has a specific smooth behavior at x that showcases a gradual, non-abrupt change.

• The function must be smooth around the point of interest.
• The derivative sign change provides insight into the zero's multiplicity.
• Differentiability ensures the ability to evaluate and analyze the function's behavior continuously up to n derivatives.

Continuity

Continuity is another vital concept that tends to be interconnected with differentiability. For a function to be continuous at a point, it must meet a few conditions: it must be defined at the point, the limit of the function as it approaches the point from both directions must exist, and the limit must equal the function's value at the point.

In the context of zeros of multiplicity, the surrounding function, particularly a component like g(x) in our problem, must be continuous. This requirement ensures that as you approach the zero, the function doesn’t jump or become undefined. Instead, it should display consistent and predictable behavior, which is crucial for understanding the nature of zeros.

• Continuity ensures there are no disruptions in the function's value around the zero.
• It guarantees that the properties of the functions like g(x) hold as x approaches x0.
• Smooth transitions in the function’s behavior help in analyzing and confirming the zero's multiplicity.

Zero of a Function

A zero of a function, quite simply, is a value for which the function equals zero. However, zeros may differ in terms of their multiplicity, which describes how many times a particular solution appears. This multiplicity denotes the number of times the function touches or crosses the x-axis at that point.

In the given problem, a zero of multiplicity n means that not only does the function equal zero at x0, but its first (n-1) derivatives also equal zero at x0. The nth derivative should not be zero; it holds a non-zero value that signifies the function moves away from the x-axis after touching or tangentially touching it n times.

• The zero's multiplicity describes its behavior at the x-axis.
• A higher multiplicity often suggests that the zero represents a higher order root.
• Multiplicity is essential to determine how the function behaves immediately around the zero.