Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Suppose that \(f\) is a continuous function and that \(f(3)=2, f^{\prime}(3)=0\), and \(f^{\prime \prime}(3)=3\). At \(x=3\), does \(f\) have a local maximum, a local minimum, neither a local maximum nor a local minimum, or is it impossible to determine? Explain your answer.

Step-by-Step Solution

Answer

The function \(f\) has a local minimum at \(x=3\).

Step 1: Evaluate the First Derivative at \(x=3\)

According to the information given, we know that \(f^{\prime}(3)=0\). The first derivative at a point being equal to zero indicates that the function could have a local maximum, a local minimum, or neither at the point (it is a critical point).

Step 2: Evaluate the Second Derivative at \(x=3\)

We are also given that \(f^{\prime\prime}(3)=3\), which is greater than zero. According to the second derivative test, if the second derivative at a point is positive, then the function has a local minimum at that point. Consequently, this tells us that the function \(f\) has a local minimum at \(x=3\) since the second derivative is greater than zero.

Key Concepts

Second Derivative Test

The Second Derivative Test is a powerful tool used in calculus to determine the nature of critical points in a function. When we find a point where the first derivative of a function is zero, known as a critical point, we use the second derivative to determine what type of extremum we have at that point.

• If the second derivative, \( f''(x) \), is positive at a critical point, the function is concave up, suggesting a local minimum.
• If \( f''(x) \) is negative, the function is concave down, indicating a local maximum.
• If \( f''(x) \) equals zero, the test is inconclusive.

For the exercise given, since \( f''(3) = 3 \) which is greater than zero, it reveals that there is a local minimum at \( x = 3 \). This helps to quickly identify the local behavior of the function.

Local Minimum

A local minimum of a function occurs at a point where the function value is lower than or equal to the function values at points in its immediate vicinity. When using the Second Derivative Test, if the second derivative is positive, it assures us that the function has this kind of dip at that point.
In the context of the exercise, since \( f''(3) > 0 \), \( x = 3 \) is a local minimum, meaning that the function's graph dips down to this point before rising again. This is an essential concept for analyzing graphs and finding the optimal solutions in many real-world applications, from economics to physics.

Continuous Function

A continuous function is a function without any interruptions, jumps, or holes in its domain. Mathematically, a function \( f(x) \) is continuous at a point \( x = a \) if the following three conditions are met:

• \( f(a) \) is defined.
• \( \lim_{x \to a} f(x) = f(a) \).
• The limit \( \lim_{x \to a} f(x) \) exists.

In practical terms, you can draw the graph of a continuous function without lifting your pen from the paper. In the exercise, the continuity of the function on which the test is performed ensures that there are no sudden changes in behavior around \( x=3 \), allowing us to rely on derivative tests to find local extrema confidently.

Critical Point

A critical point of a function occurs where its derivative is zero or undefined. This is where the function might have a local maximum, a local minimum, or neither.
When analyzing a function, spotting these critical points is crucial as they can reveal tips or turns within a function's graph. In our exercise, since \( f'(3) = 0 \), \( x = 3 \) is identified as a critical point, representing a potential for a peak or trough in the function's graph.
To determine the nature of this critical point, we then employed the Second Derivative Test, finding out that the function has a local minimum at this point. Hence, a critical point provides the location, while the second derivative test reveals the nature—making them both integral to understanding the function's behavior.