Problem 81
Question
Suppose that the point \((a, f(a))\) is a point of inflection of the graph of \(y=f(x)\). Prove that the number \(a\) gives rise to a relative extremum of the function \(f^{\prime}\).
Step-by-Step Solution
Verified Answer
To prove that \(a\) gives rise to a relative extremum for the function \(f'\) given that \((a, f(a))\) is a point of inflection on the graph of \(y = f(x)\), we analyzed the first and second derivatives. At a point of inflection, the second derivative changes its sign, and by applying the Mean Value Theorem, we showed that \(f'(a)\) is either a relative maximum or minimum. Thus, \(a\) leads to a relative extremum for \(f'\).
1Step 1: Find the second derivative of the function
Since we are given that \((a, f(a))\) is a point of inflection on the graph of \(y = f(x)\), we will first find the second derivative of the function \(f(x)\), denoted as \(f''(x)\).
2Step 2: Analyze the sign change of the second derivative
We know that at a point of inflection, the second derivative \(f''(x)\) changes its sign. Let's assume that \(f''(a - h) > 0\) for some \(h > 0\) and \(f''(a + h) < 0\). Now, according to the Mean Value Theorem, there exists a point \(c \in (a - h, a + h)\) such that \(f'(c) - f'(a - h) = (c - (a - h))f''(d)\) for some \(d \in (a - h, c)\).
Since, \(f''(d) > 0\) and \(c - (a - h) > 0\), we have \(f'(c) > f'(a - h)\).
Similarly, there exists a point \(\tilde{c} \in (a - h, a + h)\) such that \(f'(a + h) - f'(\tilde{c}) = ((a + h) - \tilde{c})f''(\tilde{d})\) for some \(\tilde{d} \in (\tilde{c}, a + h)\).
Since, \(f''(\tilde{d}) < 0\) and \((a + h) - \tilde{c} > 0\), we have \(f'(a + h) > f'(\tilde{c})\).
Clearly, \(f'(a - h) < f'(c) > f'(a + h)\). Thus, \(f'(c)\) is a relative maximum of the function \(f'\) and \(c=a\).
We can also prove this in the case when \(f''(a - h) < 0\) and \(f''(a + h) > 0\). In that case, \(f'(c)\) will be a relative minimum.
3Step 3: Conclude the proof
We have shown that \(a\) gives rise to a relative extremum of the function \(f'(x)\), given that \((a, f(a))\) is a point of inflection on the graph of \(y = f(x)\). This completes the proof.
Key Concepts
second derivativerelative extremumMean Value Theorem
second derivative
The second derivative of a function, denoted as \( f''(x) \), is a critical tool in calculus for understanding the concavity of a function. When we take the first derivative \( f'(x) \), we get the slope or rate of change of the function; with the second derivative, we analyze how that rate of change itself is changing.
In the context of identifying inflection points, which occur where the curve changes concavity, the second derivative plays a pivotal role. An inflection point is characterized by a change in the sign of \( f''(x) \). This means that \( f''(x) \) will be positive on one side of the inflection point and negative on the other.
At such points, the graph may change from being concave up (like a cup: \( f''(x) > 0 \)) to concave down (like a cap: \( f''(x) < 0 \)), or vice versa. This change indicates that the acceleration of the slope is inverting, contributing crucially to the study of the function's shape.
In the context of identifying inflection points, which occur where the curve changes concavity, the second derivative plays a pivotal role. An inflection point is characterized by a change in the sign of \( f''(x) \). This means that \( f''(x) \) will be positive on one side of the inflection point and negative on the other.
At such points, the graph may change from being concave up (like a cup: \( f''(x) > 0 \)) to concave down (like a cap: \( f''(x) < 0 \)), or vice versa. This change indicates that the acceleration of the slope is inverting, contributing crucially to the study of the function's shape.
relative extremum
A function's relative extremum refers to the relative maxima or minima within a particular interval. For a function \( f(x) \), a point \( c \) is a relative maximum if \( f(c) \) is greater than the function values nearby, that is, in some small interval around \( c \). Conversely, \( c \) is a relative minimum if \( f(c) \) is less than those nearby values.
Connecting this to the second derivative, the concept of a relative extremum helps us understand how \( f'(x) \), the first derivative, behaves. When \( f'(x) \) changes from increasing to decreasing, it often indicates a relative maximum. If it changes from decreasing to increasing, it suggests a relative minimum.
In the given exercise, since a point of inflection can lead to \( f'(x) \) having either a maximum or minimum at that point, analyzing the behavior of \( f''(x) \) is crucial. Where the second derivative changes sign, \( f' \) could exhibit a relative extremum, indicating how the function's slope peaks or bottoms out.
Connecting this to the second derivative, the concept of a relative extremum helps us understand how \( f'(x) \), the first derivative, behaves. When \( f'(x) \) changes from increasing to decreasing, it often indicates a relative maximum. If it changes from decreasing to increasing, it suggests a relative minimum.
In the given exercise, since a point of inflection can lead to \( f'(x) \) having either a maximum or minimum at that point, analyzing the behavior of \( f''(x) \) is crucial. Where the second derivative changes sign, \( f' \) could exhibit a relative extremum, indicating how the function's slope peaks or bottoms out.
Mean Value Theorem
The Mean Value Theorem (MVT) is a fascinating result from calculus with profound implications for the behavior of derivates over an interval. According to the MVT, for a function \( f(x) \) that is continuous over \([a, b]\) and differentiable over \((a, b)\), there exists at least one point \( c \) in \((a, b)\) such that \[ f'(c) = \frac{f(b) - f(a)}{b - a} \] This expression states that there is at least one point where the instantaneous slope (or derivative) of the function matches the average slope between those two points.
The theorem underpins many results in analysis and as demonstrated in our exercise, plays a critical role in analyzing the change of rates within intervals. During the process wherein \( f'(x) \) becomes a relative extremum, the theorem guarantees that \( f'(x) \) will equate to the average slope value computed over an interval where \( f''(x) \) changes sign.
This signifies that not only does \( f' \) chart relative maximums or minimums, but also correlates to a directly measurable change in the function's first derivative over that interval. Hence, MVT offers a bridge between the derivatives' theoretical behavior and their observable consequences.
The theorem underpins many results in analysis and as demonstrated in our exercise, plays a critical role in analyzing the change of rates within intervals. During the process wherein \( f'(x) \) becomes a relative extremum, the theorem guarantees that \( f'(x) \) will equate to the average slope value computed over an interval where \( f''(x) \) changes sign.
This signifies that not only does \( f' \) chart relative maximums or minimums, but also correlates to a directly measurable change in the function's first derivative over that interval. Hence, MVT offers a bridge between the derivatives' theoretical behavior and their observable consequences.
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