Problem 22 - Calculus with Concepts in Calculus

Question

Determine all functions \(f\) satisfying the given conditions. $$ f^{\prime \prime}(x)=0, f^{\prime}(-2)=1 $$

Step-by-Step Solution

Answer

The functions are of the form \( f(x) = x + d \), where \( d \) is any constant.

Step 1: Understanding the Second Derivative

The condition \( f''(x) = 0 \) tells us that the second derivative of the function is zero for all \( x \). This means that the first derivative, \( f'(x) \), must be a constant function. Let this constant be \( c \). Therefore, \( f'(x) = c \).

Step 2: Integrate to Find the First Derivative

Since \( f'(x) = c \), integrating this with respect to \( x \) gives us the function \( f(x) = cx + d \), where \( d \) is the constant of integration.

Step 3: Apply Initial Condition for First Derivative

We are given that \( f'(-2) = 1 \). Using \( f'(x) = c \), we substitute \( x = -2 \) and set it equal to 1: \( c = 1 \). Therefore, \( f'(x) = 1 \).

Step 4: Find the Function \( f(x) \)

Using \( f'(x) = 1 \), we integrate again to find \( f(x) = 1x + d = x + d \). The function is \( f(x) = x + d \), where \( d \) is any constant.

Key Concepts

Second Derivative

The second derivative of a function, denoted as \( f''(x) \), provides information about the curvature of the graph of a function. When we say \( f''(x) = 0 \), it indicates that the graph does not curve at all, meaning it is a straight line. This is because the rate of change of the slope (which the second derivative measures) is zero, signifying no curvature. Thus, the function itself is linear. This condition is crucial as it simplifies the process of finding the function \( f(x) \) by reducing our task to evaluating the constant first derivative.

Integration

Integration is the process of finding a function from its derivative, essentially the reverse of differentiation. When we integrate the first derivative \( f'(x) = c \), we find the original function \( f(x) \). In our scenario, \( f'(x) \) is a constant, so integrating \( c \) with respect to \( x \) yields the function \( f(x) = cx + d \).
This integration introduces a constant of integration \( d \), which accounts for the indefinite nature of the integral.
It's important to remember that any function with the same derivative differs only by a constant, thus necessitating the constant of integration.

Constant of Integration

The constant of integration arises when we perform indefinite integration. In the process of finding \( f(x) \) from \( f'(x) \), we add a constant \( d \). This constant represents the family of functions differing by a vertical shift, each having the same derivative.
In our specific problem, \( f'(x) = 1 \) leads to \( f(x) = x + d \). Without further information, \( d \) remains arbitrary. However, additional conditions can help determine \( d \) if provided.

• The constant ensures all possible solutions from integration are covered.
• Allows for the function to match given conditions or initial values.

Derivative

A derivative, denoted as \( f'(x) \), describes how a function changes at any given point. It captures the slope or rate of change of the function. In our exercise, we initially know the second derivative \( f''(x) \) is zero, indicating a constant first derivative \( f'(x) = c \).
The given condition \( f'(-2) = 1 \) determined the value of \( c \) to be 1. Therefore, deriving the next function involved simply substituting \( c \) to find \( f'(x) = 1 \).
Understanding derivatives is crucial because they provide insights not just into the behavior of functions but also into constructing solutions that meet specific criteria.