Problem 20 - Calculus for Biology and Medicine

Question

In Problems , 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

Start by finding the derivative of the function \( f(x) = (x-4)^2 \). Use the power rule: \( \frac{d}{dx}(x^n) = n \cdot x^{n-1} \). The derivative is \( f'(x) = 2(x-4) \).

Step 2: Solve for f'(c) = 0

Set the derivative equal to zero to find the critical points: \( 2(x-4) = 0 \). Solve for \( x \) to find \( x = 4 \). Thus, \( c = 4 \).

Step 3: Determine the Sign of the Second Derivative

Compute the second derivative to determine the concavity of \( f(x) \) at \( x = 4 \). The second derivative is \( f''(x) = 2 \). Since \( f''(x) > 0 \), the function is concave up.

Step 4: Interpret the Results

Since the second derivative is positive at \( x = c = 4 \), \( f(x) \) has a local minimum at \( x = 4 \). This is because the function is concave up, indicating a minimum point.

Key Concepts

Derivative

To determine the critical points of a function, we first need to calculate its derivative. The derivative provides the rate at which the function's value changes with respect to its independent variable, typically denoted as \( x \). It is a fundamental concept that aids in identifying behaviors like slopes and turning points. In the example function \( f(x) = (x-4)^2 \), calculating the derivative involves applying the power rule, which will be explained later.

The derivative of \( f(x) \), expressed as \( f'(x) \), is crucial because setting it equal to zero gives us the potential locations of local maxima or minima. This position is called a critical point. By solving \( f'(x) = 0 \), we can identify these critical values that potentially indicate where the function changes direction from increasing to decreasing, or vice versa.

Local Extremum

A local extremum refers to points on the graph of a function where it reaches a minimum or maximum within a certain interval. When such points are identified using the derivative, we verify their nature by further analysis. For instance, in situations where the first derivative equals zero, the function might have a local extremum at those \( x \)-values.

To determine whether it is indeed a local minimum or maximum, the second derivative test can be utilized. If the second derivative \( f''(x) \) at the critical point \( c \) is positive, the function is concave up and there is a local minimum at \( x = c \). Conversely, if \( f''(c) \) is negative, it implies a concave down graph at that point and indicates a local maximum. In the given function, \( f''(x) = 2 \) which is positive, meaning there is a local minimum at \( x = 4 \).

Power Rule

The power rule is a simple yet powerful tool used to find the derivative of functions in the form of \( x^n \). When you apply the power rule, you bring down the exponent as a coefficient and subtract one from the exponent. Here's the formula:

• \( \frac{d}{dx}(x^n) = n \cdot x^{n-1} \)

For example, using the function \( f(x) = (x-4)^2 \), the power rule allows us to easily calculate \( f'(x) \). We recognize that \((x-4)^2 = ((x-4))^2\) fits the form, where \( n = 2 \), so our derivative would be \( 2 \cdot ((x-4))^1 = 2(x-4) \).

The simplicity of the power rule makes it an indispensable method for differentiating polynomial functions. It's effective for both constant and variable terms and is central to calculus operations involving rate of change analysis.