Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(H(x)\) have three continuous derivatives, and be such that \(H(1)=H^{\prime}(1)=H^{\prime \prime}(1)=0\), but \(H^{\prime \prime \prime}(1) \neq 0 .\) Does \(H(x)\) have a local maximum, local minimum, or a point of inflection at \(x=1\) ? Justify your answer

Step-by-Step Solution

Answer

\(x=1\) is a point of inflection.

Step 1: Understand the Given Information

We are given that the function \( H(x) \) and its first two derivatives \( H'(x) \) and \( H''(x) \) at \( x=1 \) are all equal to zero: \( H(1) = 0 \), \( H'(1) = 0 \), and \( H''(1) = 0 \). The third derivative at this point is not zero: \( H'''(1) eq 0 \). This is critical information for determining the nature of the point at \( x=1 \).

Step 2: Apply the Second Derivative Test

The second derivative test typically helps in determining local maxima or minima. It states that if \( H''(x) = 0 \), we need to further analyze the derivatives. However, since \( H''(1) = 0 \), this directly tells us that the second derivative test is inconclusive at \( x=1 \).

Step 3: Analyze Higher Derivatives

If the second derivative test is inconclusive, examine higher derivatives. Since \( H''(1) = 0 \) and \( H'''(1) eq 0 \), this implies that \( H(x) \) does not show a typical local maximum or minimum behavior here. Specifically, since the first non-zero higher derivative is of odd order, \( x=1 \) must be a point of inflection.

Key Concepts

Calculus

Calculus helps us understand how a function behaves by examining its derivatives. Derivatives provide information about the function's rate of change at any given point.

• The first derivative, often noted as \( H'(x) \), tells us how the function itself is changing. It can indicate where the function reaches its peak (local maximum) or its lowest value (local minimum).
• The second derivative, \( H''(x) \), gives insight into how the rate of change itself is changing, which helps in identifying points of acceleration or deceleration.

In this particular problem, we look at up to the third derivative, \( H'''(x) \), to determine the nature of the point at \( x=1 \). When the second derivative is zero, the third derivative helps clarify the situation further.

Derivative Test

The derivative test is a common technique in calculus to determine the nature of critical points of a function using its derivatives.

• First Derivative Test: If \( H'(x) = 0 \) at a point, this indicates a potential local maximum or minimum, known as a critical point.
• Second Derivative Test: If \( H''(x) > 0 \) at a critical point, the function has a local minimum. If \( H''(x) < 0 \), it has a local maximum. If \( H''(x) = 0 \), the second derivative test is inconclusive, and we cannot determine local extremum directly.

Since \( H''(1) = 0 \), the second derivative test doesn't give us a clear answer about whether \( x=1 \) is a max, min, or neither. This is why exploring further derivatives, such as the third derivative, is essential.

Third Derivative

The third derivative of a function, \( H'''(x) \), can provide further insights when the first and second derivatives are zero at a point.In our exercise, the third derivative, \( H'''(1) eq 0 \), plays a pivotal role. Since the first and second derivatives at \( x=1 \) are zero, we know there's no immediate indication of maxima or minima. Instead, the non-zero third derivative signals a change in concavity at this point.

• If the third derivative at a point is not zero, and the first two derivatives are zero, it implies a point of inflection at that point.
• The sign of the third derivative doesn't affect this conclusion—it just confirms the transition from concave up to concave down or vice versa.

Thus, recognizing that \( H'''(1) eq 0 \), we confidently call \( x=1 \) a point of inflection.

Local Extrema

Local extrema refer to the highest or lowest points in a small section of the function’s graph, known as local maximum or minimum. Identifying local extrema typically involves using the first derivative test to find critical points (where \( H'(x) = 0 \)) and the second derivative test to confirm the nature of these critical points (as either max or min).When both these tests fail, as in our case, where \( H''(1)=0 \), we look to higher-order derivatives like the third derivative. Unfortunately, with \( H'''(1) eq 0 \), we can only assert the presence of a point of inflection at \( x=1 \), not a local maximum or minimum.To summarize:

• Local maxima and minima are where the function hits the top or bottom locally.
• If higher-order derivatives are non-zero, they might not signify extrema but only indicate changing concavity, leading to a point of inflection.