Problem 53 - 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: Analyze the derivative information

We know that the derivative of the function, \(f^{\prime}(x)\), is zero for all \(x\) in \(\mathbb{R}\). This means the function \(f(x)\) has a constant slope of zero. A function with a zero derivative is constant.

Step 2: Identify the characteristic of a constant function

Since the derivative is always zero, \(f(x)\) must be constant for all \(x\). A constant function can be expressed as \(f(x) = C\), where \(C\) is a constant real number.

Step 3: Determine the value of the constant

We use the given condition \(f(2) = 3\) to find the constant \(C\). Since \(f(x)\) is constant and \(f(2) = 3\), it follows that \(C = 3\). Therefore, \(f(x) = 3\) for all \(x\).

Key Concepts

Constant Functions

In mathematics, a constant function is a unique and simple concept. A constant function is one that returns the same value regardless of the input. For example, if a function is defined as \( f(x) = C \), where \( C \) is a constant, then for every value of \( x \), \( f(x) \) will always equal \( C \).
Constant functions have some interesting properties:

• Their graphs are horizontal lines in the Cartesian plane.
• Their slope is zero, meaning there is no vertical change as you move along the graph.

This specific type of function is especially important in calculus because it represents an intuitive case where the rate of change is consistently zero. No matter how much \( x \) changes, the output remains steady, encapsulating the ‘constant’ aspect.

Zero Derivative

A derivative reflects the rate at which a function is changing at any point, and it's a fundamental idea in calculus. When the derivative \( f'(x) = 0 \) for all \( x \), this tells us something meaningful about the function itself—it doesn't change.
If we picture a derivative as the slope, a zero derivative signifies that we have a perfectly flat slope that doesn't incline or decline. In terms of the function:

• \( f'(x) = 0 \) indicates unchanging behavior.
• The function remains invariant, or constant, across its domain.

In practical terms, a zero derivative often simplifies calculations in problem-solving scenarios by immediately noting the status of a function as constant. This quality helps bypass the need for complex examination of function behavior over intervals, thus making calculus more manageable.

Calculus Problem Solving

Solving calculus problems involves identifying and understanding derivatives and their implications. In our exercise, we needed to determine a function based on its derivative always being zero. Recognizing the zero derivative, we deduced that the function must be constant.
Effective calculus problem solving aligns with these steps:

• Understand the Derivative Information: Recognize what a zero derivative implies about the function.
• Use Given Conditions: Apply any known values, like \( f(2) = 3 \), to identify constants within the solution.
• Verify: Ensure the solution remains consistent by checking the function's defined behavior across its range.

By logically following these steps, complex problems become digestible, ultimately adding clarity and precision to the process of solving calculus exercises.