Problem 28 - Calculus with Concepts in Calculus

Question

Determine all functions \(f\) satisfying the given conditions. $$ f^{(n)}(x)=0 \text { for any positive integer } n $$

Step-by-Step Solution

Answer

The function \(f(x)\) is a constant, \(f(x) = C\).

Step 1: Understand the Concept of Derivative

The notation \(f^{(n)}(x)\) represents the \(n\)-th derivative of the function \(f(x)\). If a function's \(n\)-th derivative is zero, it means that all higher-order derivatives are also zero.

Step 2: Analyze the Condition for Zero Derivatives

Given that \(f^{(n)}(x) = 0\) for any positive integer \(n\), this implies all derivatives from the \(n\)-th onward are zero. Hence, the function \(f(x)\) must be a polynomial of degree less than \(n\).

Step 3: Deduce the Degree of the Polynomial

If we have \(f^{(n)}(x) = 0\) for any \(n\), then \(f(x)\) must be a constant function. This is because, for any polynomial of degree \(d\), the \((d+1)\)-th derivative and higher are zero. Consequently, the only real solutions that will satisfy the condition for all \(n\) are constant functions.

Step 4: Find the General Solution

Based on the analysis, the function \(f(x)\) that satisfies \(f^{(n)}(x) = 0\) for any positive integer \(n\) can only be a constant function. Therefore, the general solution is \(f(x) = C\), where \(C\) is a constant.

Key Concepts

Understanding Derivatives

In calculus, derivatives play a crucial role in understanding how functions change. Essentially, the derivative measures the rate at which a function is changing at any given point. Consider it as the slope or the 'steepness' of a function's graph at a point.

Notation like \(f'(x)\), \(f''(x)\), and so forth up to \(f^{(n)}(x)\) helps to denote the first, second, and \(n\)-th derivatives of a function respectively. When the \(n\)-th derivative of a function \(f(x)\) equals zero, it implies that from this point, the function's behavior no longer changes significantly. All subsequent higher-order derivatives will also be zero, indicating a stagnation in the 'motion' of the graph after the \(n\)-th derivative.

In simpler terms, if you keep taking derivatives of a function and eventually every derivative you find is zero, the function has reached a basic, unchanging form.

Exploring Polynomials

Polynomials are one of the simplest types of functions in calculus, structured as the sum of terms of the form \(ax^n\), where \(a\) is a coefficient and \(n\) is a non-negative integer representing the degree of the term. The degree of a polynomial is the highest exponent \(n\) which appears in the polynomial. This degree dictates the 'shape' and certain behaviors of the polynomial.

For example, a quadratic polynomial like \(ax^2 + bx + c\) has a degree of 2, while a linear polynomial \(mx + b\) has a degree of 1. As we differentiate polynomials, each differentiation reduces the degree by one until it becomes zero.

• The first derivative of a polynomial lowers its degree by 1.
• If a polynomial of degree \(d\) is differentiated \((d+1)\) times, we obtain a zero function.

This realization helps us understand that once all derivatives of a polynomial become zero, the polynomial itself becomes a simpler form, specifically a constant.

Constant Functions

Constant functions are the simplest type of functions in calculus, where the function always returns the same value, regardless of the input. The graph of a constant function is a horizontal line.

• Mathematically, a constant function can be represented as \(f(x) = C\), where \(C\) is a real number.
• All derivatives of a constant function are zero — this is because there's no change or slope in a flat line.

In the context of the problem, if all derivatives \(f^{(n)}(x) = 0\) for any \(n\), it indicates that the function is unchanging, which aligns with the definition of a constant function. This simplicity makes constant functions a fundamental concept in understanding how differentiated forms of more complex equations stabilize at a level that shows no change.
Therefore, when analyzing derivatives that become zero, it quickly leads to the conclusion that the original function must be a constant.