Problem 20 - Calculus for Biology and Medicine

Question

Find \(c\) such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) \(f(x)=(x-4)^{2}\)

Step-by-Step Solution

Answer

The function has a local minimum at \( x = 4 \).

Step 1: Find the Derivative

First, we need to find the derivative of the function \( f(x) = (x - 4)^2 \). Using the power rule, the derivative is: \( f'(x) = 2(x - 4) \).

Step 2: Solve for Critical Points

To find the value of \( c \) where \( f'(c) = 0 \), set the derivative equal to zero and solve for \( x \): \[ 2(x - 4) = 0 \]. Dividing both sides by 2 gives \( x - 4 = 0 \). Adding 4 to both sides, we get \( x = 4 \). Thus, \( c = 4 \).

Step 3: Determine if Local Extremum Exists

To determine if there is a local extremum at \( x = c \), we check the second derivative, \( f''(x) \). The second derivative of \( f(x) = (x - 4)^2 \) is \( f''(x) = 2 \), which is positive, indicating that \( f(x) \) is concave up at \( x = 4 \). This means \( f(x) \) has a local minimum at \( x = 4 \).

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus. They measure how a function changes as its input changes. Think of a derivative as the 'speed' at which a function's value changes. For example, the derivative of a position function with respect to time gives the velocity. In our exercise, we dealt with

• a specific quadratic function: \(f(x) = (x-4)^2\).

To find how quickly this function changes, we calculated its derivative. The process involved identifying how the output changes with respect to the input \(x\). This is a powerful tool because knowing the derivative can tell you many things about the function's behavior at different points.

Exploring Local Extrema

A local extremum is a point on a function where it achieves a maximum or minimum value in a small interval around that point. It's important because these points often represent optimal values in real-world contexts, such as the highest profit or the least cost.
In mathematical terms, a function has a local minimum at a point \(x = c\) if \(f(c)\) is the lowest point in the neighborhood around \(c\). Conversely, it's a local maximum if \(f(c)\) is the highest. To determine if the point is a local extremum, we:

• First find where the derivative \(f'(x)\) equals zero, as this often indicates where extreme values might occur.
• Then check the concavity of the function by analyzing the second derivative. In our case, a positive second derivative (\(f''(x) = 2\)) signifies that the function is concave up, indicating a local minimum at that point.

The Power Rule in Calculus

The power rule is one of the simplest and most commonly used methods for finding derivatives. It states that if you have a function such as \(x^n\), the derivative is \(nx^{n-1}\). This rule helps simplify the process of differentiation, especially for polynomials.
In our example:

• We started with \(f(x) = (x-4)^2\).
• Applying the power rule, we treated \((x-4)\) as a single term to easily find the derivative \(f'(x) = 2(x-4)\).

This quick and efficient method of taking derivatives is invaluable for tackling more complex calculus problems. Mastery of the power rule is essential for anyone studying calculus as it plays a crucial role in almost every derivation, allowing students to focus on more intricate parts of problems.