Problem 26 - 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 \(x=c .\) $$ f(x)=(x+1)^{3} $$

Step-by-Step Solution

Answer

The critical point is at \( x = -1 \), with the local extremum test being inconclusive.

Step 1: Find the Derivative

First, we need to find the derivative of the function \( f(x) = (x+1)^3 \). Using the power rule, the derivative is \( f'(x) = 3(x+1)^2 \).

Step 2: Set the Derivative to Zero

To find critical points, set the derivative equal to zero: \( 3(x+1)^2 = 0 \). Simplify to get \((x+1)^2 = 0\).

Step 3: Solve for Critical Points

Solve the equation \((x+1)^2 = 0\) to find \( x+1 = 0 \). This gives \( x = -1 \). Thus, \( c = -1 \).

Step 4: Determine the Nature of Critical Points

Use the second derivative test to determine if there is a local extremum. First, find the second derivative: \( f''(x) = 6(x+1) \). Evaluate it at \( x = -1 \): \( f''(-1) = 6(-1+1) = 0 \). Since the second derivative is zero, the test is inconclusive.

Key Concepts

Understanding Derivatives

The derivative of a function is a fundamental concept in calculus. It represents how a function changes as its input changes. You can think of it as the slope of the tangent line to the function at any given point.
To find the derivative, you often use basic differentiation rules, such as the power rule, product rule, or quotient rule, depending on the form of the function.

• The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \).

For example, in the exercise provided, for the function \( f(x) = (x+1)^3 \), the power rule is applied to find that \( f'(x) = 3(x+1)^2 \). This derivative tells us how the function \( f(x) \) behaves around any point \( x \). If you set this derivative equal to zero, you can identify important points, specifically where the function might have a local extremum.

Identifying Local Extrema

A local extremum is a point where a function reaches a local maximum or minimum. Identifying these points is crucial as they tell us where the function reaches the highest or lowest values in a small interval.
To find a local extremum, first identify the critical points by setting the derivative equal to zero. If the derivative is zero at a point, the slope of the function at that point is flat, meaning potential maxima or minima occur.

• For \( f(x) = (x+1)^3 \), setting the derivative \( f'(x) = 3(x+1)^2 \) equal to zero leads to the critical point \( x = -1 \), meaning the function is flat at this point.

Further analysis is required to determine the nature of this extremum, which can be done using the second derivative test.

The Second Derivative Test

The second derivative test is a handy method to determine the nature of critical points. This test uses the second derivative of a function, which tells us how the slope of the function is changing.

• If the second derivative \( f''(x) \) is positive at a critical point \( x = c \), the function has a local minimum.
• If \( f''(x) \) is negative, there's a local maximum.
• If \( f''(x) = 0 \), the test is inconclusive and the nature of the critical point must be determined using other methods, such as analyzing higher order derivatives or the behavior of the original function tested over intervals.

In the exercise example, for \( f(x) = (x+1)^3 \), the second derivative comes out to \( f''(x) = 6(x+1) \). When checking at \( x = -1 \), the result is zero, indicating that this test cannot confirm if \(-1\) is a point of local extremum.