Problem 54 - Calculus for Biology and Medicine

Question

Suppose that \(f\) is differentiable for all \(x \in \mathbf{R}\) with \(f(2)=3\) and \(f^{\prime}(x)=0\) for all \(x \in \mathbf{R}\). Find \(f(x)\).

Step-by-Step Solution

Answer

The function is constant: \( f(x) = 3 \) for all \( x \).

Step 1: Understand the Problem

We are given a function \( f \) that is differentiable everywhere, and its derivative, \( f'(x) \), is always 0 for every real number \( x \). We also know that \( f(2) = 3 \). We need to find the form of \( f(x) \).

Step 2: Analyzing the Derivative Condition

The derivative \( f'(x) = 0 \) for all \( x \) implies that \( f(x) \) is a constant function. This is because the derivative represents the rate of change, and a zero derivative indicates no change everywhere across the domain.

Step 3: Use the Initial Condition

Since \( f(x) \) is a constant function, it takes the same value at every point \( x \). We use the given condition \( f(2) = 3 \) to determine the constant value of the function. Thus, \( f(x) = 3 \) for all \( x \).

Step 4: Express the Solution

Having determined that \( f(x) \) is constant and must meet the initial condition, we conclude that \( f(x) = 3 \) for every real number \( x \). This satisfies both the condition that the derivative is zero and \( f(2) = 3 \).

Key Concepts

Understanding Derivatives

A derivative provides crucial information about how a function changes. In simpler terms, it measures the rate at which the function's output value changes as its input value changes. More specifically, for a function \( f(x) \), the derivative, denoted by \( f'(x) \), tells us the slope of the tangent line to the curve at any point \( x \). For example, if the slope is positive, the function is increasing at that point, whereas a negative slope indicates a decrease.

• If \( f'(x) = 0 \) for all \( x \), the function does not change as \( x \) changes. This means the function is completely flat everywhere across the domain.
• This is the exact scenario in the problem you're working with, where \( f'(x) = 0 \) for all \( x \). This directly leads us to conclude that \( f(x) \) does not vary—it is a constant function.

Constant Functions Simplified

Constant functions are among the most straightforward types of functions. A constant function takes the same value no matter what input you provide. It's like a flat, horizontal line on a graph.

• The equation for a constant function looks like \( f(x) = c \), where \( c \) is a constant number.
• In this scenario, where the derivative \( f'(x) = 0 \), we are guaranteed a constant function because no change in \( y \) occurs as \( x \) changes.
• Imagine drawing a horizontal line on a graph—it doesn't go up, it doesn't go down. It simply stays at the same level continuously.

Knowing that \( f(2) = 3 \) helps identify that constant value \( c \). Hence, in this particular problem, \( f(x) = 3 \) for all \( x \).

Role of the Initial Condition

An initial condition in mathematics is a crucial piece of information that allows us to determine an unknown constant within equations. It acts like a starting point or a specific point through which the function must pass.

• In the given problem, the initial condition \( f(2) = 3 \) helps establish the value of the constant function \( f(x) \).
• Without this information, knowing \( f'(x) = 0 \) would only tell us that \( f(x) \) is constant, but not what that constant value actually is.
• This initial condition pins down \( f(x) \) to equal 3 for every \( x \), giving us the complete solution.

Hence, combining the derivative information and the initial condition, we've derived that our function, \( f(x) = 3 \), meets all the given criteria in the problem.