Problem 71 - Calculus of a Single Variable

Question

Analyzing a Critical Number \(A\) differentiable function \(f\) has one critical number at \(x=5\) . Identify the relative extrema of \(f\) at the critical number when \(f^{\prime}(4)=-2.5\) and \(f^{\prime}(6)=3\)

Step-by-Step Solution

Answer

The function \(f\) has a relative minimum at \(x=5\).

Step 1: Understand the Problem

The function \(f\) is given to be differentiable and has a critical number at \(x=5\), which means \(f^{\prime}(5)=0\). Since the value of the derivative of \(f\) at \(x=4\) is given as \(-2.5\) which is negative and the value of the derivative at \(x=6\) is given as \(3\) which is positive, one needs to determine whether or not there is a change in the sign of the derivative around the critical number. For this, both the intervals \((-\infty,5)\) and \((5,\infty)\) will be analyzed.

Step 2: Apply the First Derivative Test to Check for a Sign Change

The first derivative is negative on the interval \((-\infty,5)\) and positive on (5,\infty). This indicates the function \(f\) goes from decreasing to increasing as \(x\) passes through 5.

Step 3: Identifying the relative Extrema

Because the function \(f\) Goes from going downward (negative slope) to going upward (positive slope) at \(x=5\), according to the First Derivative Test, there exists a relative minimum at \(x=5\).

Key Concepts

Understanding the First Derivative Test

The First Derivative Test is a tool used in calculus to determine the nature of critical points, which are points where the derivative of a function is zero or undefined. In the context of determining relative extrema, it helps us understand the behavior of a function around these critical points.

When using the First Derivative Test, follow these steps:

• Identify critical points where the derivative is zero or does not exist.
• Analyze the sign of the derivative before and after each critical point by testing values in the intervals created around the critical points.
• If the derivative changes from negative to positive, then the critical point is a relative minimum.
• If the derivative changes from positive to negative, then the critical point is a relative maximum.
• If there is no sign change, the critical point is neither a relative maximum nor a minimum.

In our original exercise, the derivative analysis shows a negative derivative before the critical point and a positive derivative after. According to the First Derivative Test, this indicates a transition from decreasing to increasing. So, the function has a relative minimum at the critical point.

Exploring Relative Extrema

Relative extrema refer to the points where a function reaches a local maximum or minimum. These points are important for understanding the essential behavior of a function in its domain. Here’s how you can identify them:

• A relative maximum is present where the function reaches its highest value before decreasing.
• A relative minimum occurs where the function attains its lowest value before increasing.
• These can be identified using the First Derivative Test applied to the function’s critical points.

For the function in the original example, we see that at the critical point, the derivative switches from negative to positive. This indicates a transition from going down to going up, marking a relative minimum at the critical point. This understanding of relative extrema is key in many real-world applications where optimization is necessary.

Role of Differentiable Functions

Differentiable functions are functions that have derivatives at all points in their domain. The differentiability implies a smooth curve without sharp turns or cusps, ensuring that derivatives are well defined.

Differentiability is important for applying calculus techniques, such as the First Derivative Test, to analyze the function's behavior. Here's why differentiability matters:

• It guarantees that the function is smooth enough to analyze changes in its slope, which is vital for finding relative extrema.
• Differentiability ensures continuity, which is crucial when determining the intervals where a function increases or decreases.
• With a differentiable function, one can accurately calculate critical points and ascertain the nature of these points.

In our exercise, differentiability helps in identifying the precise critical point at which to conduct our test. Since it is given that the function is differentiable, we know that there are no abrupt changes, allowing us to apply the First Derivative Test with confidence.