Problem 28 - The Calculus with Analytic Geometry

Question

Suppose that \(f\) is a function for which \(f^{\prime \prime}(x)\) exists for all values of \(x\) in some open interval \(I\) and that at a number \(c\) ir I, \(f^{\prime \prime}(c)=0\) and \(f^{\prime \prime \prime}(c)\) exists and is not zero. Prove that the point \((c, f(c))\) is a point of inflection of the graph of \(f\) (HINT: The proof is similar to the proof of the second-derivative test (Theorem 5.2.1).)

Step-by-Step Solution

Answer

The point \( (c, f(c)) \) is a point of inflection because \( f''(c) = 0 \) and \( f'''(c) eq 0 \) indicate a sign change in \( f''(x) \), showing a change in concavity.

Step 1 - Understanding the basis

A point of inflection is where the concavity of a function changes. This happens if the second derivative changes sign at a point, indicating a change in concavity.

Step 2 - Consider the second derivative at point c

Given that at the point \(c\), \(f^{\prime \prime}(c) = 0\). This means that \(x = c\) is a candidate for an inflection point.

Step 3 - Analyzing the third derivative

Given that the third derivative, \(f^{\prime \prime \prime}(c)\), exists and is not zero. This suggests a change in the sign of the second derivative around \(c\).

Step 4 - Taylor series expansion

Using Taylor series expansion around the point \(c\), expand \(f^{\prime \prime}(x)\) as follows: \ f^{\prime \prime}(x) = f^{\prime \prime}(c) + (x - c)f^{\prime \prime \prime}(c)/1! + (x - c)^2f^{\prime \prime \prime \prime}(c)/2! + ... \.

Step 5 - Substitute known values

Since \(f^{\prime \prime}(c) = 0\), the expansion simplifies to: \ f^{\prime \prime}(x) = (x - c)f^{\prime \prime \prime}(c) + \text{higher-order terms} \.

Step 6 - Analyze the change in concavity

For x slightly less than c, \(f^{\prime \prime}(x) \) has the opposite sign of \(f^{\prime \prime \prime}(c) \), and for x slightly greater than c, \(f^{\prime \prime}(x) \) has the same sign as \(f^{\prime \prime \prime}(c) \). This indicates a change in concavity at \(x = c\).

Step 7 - Conclude the point of inflection

Since the concavity changes at \(x = c\), \( (c, f(c)) \) must be a point of inflection.

Key Concepts

Understanding the Second Derivative

The second derivative of a function, denoted as \( f''(x) \), measures how the rate of change of the function's slope (its first derivative) evolves.
In simpler terms, it tells us about the curvature or concavity of the function's graph.
If \( f''(x) > 0 \), the graph is concave up (shaped like a cup); if \( f''(x) < 0 \), it is concave down (shaped like a cap).

• Positive second derivative: function is concave up.
• Negative second derivative: function is concave down.

For a potential inflection point, where the function's concavity changes, the second derivative should be zero or undefined at that point.
This is why, in our exercise, the zero value of \( f''(c) \) makes \( x = c \) a candidate for an inflection point.

Role of the Third Derivative

The third derivative, denoted as \( f'''(x) \), essentially looks at the rate of change of the second derivative.
It's a higher-order derivative that helps to confirm changes in the concavity.
If \( f''(c) = 0 \) and \( f'''(c) eq 0 \), this indicates a probable change in concavity around point \( c \).
The Taylor series expansion around \( c \) helps here. By excluding higher-order terms, the Taylor expansion simplifies to:
\[ f''(x) = (x - c)f'''(c) \]
This formula indicates that the sign of \( f''(x) \) around \( x = c \) will depend entirely on the third derivative. Therefore:

• For \( x < c \), \( f''(x) \) will take the opposite sign of \( f'''(c) \)
• For \( x > c \), \( f''(x) \) will keep the same sign as \( f'''(c) \)

Hence, the presence of the non-zero third derivative confirms the change in concavity, validating the inflection point.

Understanding Concavity

Concavity refers to the direction of the curve of a function and is determined by the sign of the second derivative.

• Concave Up: A function curves upwards, resembling a 'U' shape. Here, \( f''(x) > 0 \).
• Concave Down: A function curves downwards, resembling an upside-down 'U'. Here, \( f''(x) < 0 \).

An inflection point is where the function changes its concavity.
In our example, we demonstrated that due to the third derivative being non-zero, the sign of the second derivative changes around \( x = c \). This change signifies the transition from concave up to concave down, or vice versa, confirming \( (c, f(c)) \) as an inflection point.