Question

Give an example of a function where \(f^{\prime}(x) \neq 0\) and \(f^{\prime \prime}(x)=\) 0.

Short answer

The function \(f(x) = 2x + 3\) has a first derivative \(f'(x) = 2\) (non-zero) and a second derivative \(f''(x) = 0\).

Step-by-step solution

Understanding the Given Conditions

We are looking for a function where the first derivative, \(f'(x)\), is not zero. This means the function must be either increasing or decreasing. Additionally, the second derivative, \(f''(x)\), should be zero, which implies the function has a constant rate of change, or in other words, it is a linear function.

Selecting a Suitable Function

A linear function is the simplest form that satisfies the condition \(f''(x) = 0\). Let's consider the function \(f(x) = 2x + 3\). This is a linear function with a constant slope.

Calculating the First Derivative

To ensure our function satisfies \(f'(x) eq 0\), we need to find \(f'(x)\). For the function \(f(x) = 2x + 3\), we apply the derivative formula to get \(f'(x) = 2\). Clearly, \(f'(x) eq 0\), as it is a constant value of 2.

Verifying the Second Derivative

Now, let's determine \(f''(x)\). Differentiating the first derivative, \(f'(x) = 2\), results in the second derivative \(f''(x) = 0\). This confirms the second condition of the problem.

Key concepts

Derivative

In calculus, the derivative of a function gives us valuable information about its behavior. The derivative represents the rate at which a function is changing at any given point. It is like a speedometer reading that tells us how fast or slow a function is increasing or decreasing at a specific point. Two key derivatives often discussed are the first derivative and the second derivative.

The first derivative, denoted as \( f'(x) \), indicates the rate of change of the function's value as \( x \) changes. If \( f'(x) > 0 \), the function is increasing; if \( f'(x) < 0 \), it is decreasing.

The second derivative, \( f''(x) \), provides insight into the curvature of the function. For instance, \( f''(x) = 0 \) suggests that the function is linear, and its graph is a straight line. Understanding these derivatives is crucial for analyzing the behavior and properties of functions.

Linear Function

A linear function is the simplest type of function in mathematics. It can be expressed in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This function is vital because of its constant rate of change, which makes it predictable and easy to understand.

A linear function has a graph that is a straight line, and its coefficients \( m \) and \( b \) determine the steepness and position of this line. The slope \( m \) indicates how steep the line is, and it remains the same throughout. Therefore, no matter the values of \( x \), the slope tells us how much \( f(x) \) changes with each unit of \( x \).

This uniform change is what makes linear functions distinctive. In calculus, this concept becomes especially significant as it simplifies the evaluation of derivatives, as seen in our exercise example where we found \( f'(x) = 2\).

Rate of Change

The rate of change is a fundamental concept in both mathematics and real-world applications. It tells us how one quantity changes in relation to another. In calculus, the rate of change is typically represented by the derivative. It helps determine how fast a function's value is changing at any point.

Consider a linear function, such as \( f(x) = 2x + 3 \). The rate of change of this function is constant, as indicated by its slope, \( m = 2 \). This constancy signifies that for every unit increase in \( x \), the function's value increases by 2.

The concept of rate of change becomes even more critical when dealing with real-life problems, such as speed in physics or growth rates in biology. The derivative provides a precise mathematical representation of this rate, making it an indispensable tool for interpreting and predicting behavior in natural and artificial systems.