Problem 51 - Calculus with Concepts in Calculus

Question

Show that if \(f^{*}\) exists and is continuous on an open interval containing \(c\) and if \(f\) has an inflection point at (c, \(f(c))\), then \(f^{\prime \prime}(c)=0\)

Step-by-Step Solution

Answer

If \(f\) has an inflection point at \((c, f(c))\), then \(f^{''}(c) = 0\).

Step 1: Understand the Inflection Point

An inflection point at \((c, f(c))\) means the concavity of \(f\) changes at \(x = c\). For this to happen, \(f^{''}(x)\) must change sign at \(x = c\).

Step 2: Utilize the Continuity of f''

Because \(f^{*}\) exists and is continuous around \(c\), \(f^{''}\) is defined and continuous in some neighborhood around \(c\). Therefore, \(f^{''}(x)\) doesn't have any gaps in its sign change around \(c\).

Step 3: Apply the Sign Change Criterion

Since the sign of \(f''(x)\) changes at \(x=c\), and \(f^{''}\) is continuous, it must be 0 at \(c\) to allow for a smooth transition from positive to negative or vice versa.

Step 4: Conclusion

Hence, if \(f\) has an inflection point at \((c, f(c))\), the continuity of \(f^{''}\) dictates that \(f^{''}(c) = 0\).

Key Concepts

Concavity

Concavity is a critical concept in mathematics, especially when you're analyzing the curvature of a graph. It helps us understand how the graph is bending. When a function is concave up, it means the graph resembles a U-shape. Conversely, when the graph is concave down, it looks like an upside-down U. Here's how you can remember it:

• Concave Up: Like holding a cup that can hold water.
• Concave Down: Like an upside down cup, which can't hold water.

Determining the concavity often involves using the second derivative, denoted as \(f''(x)\). If \(f''(x) > 0\), the function is concave up. If \(f''(x) < 0\), it's concave down. Understanding the nuances of concavity will help you identify key features of graphs, such as intervals of increase or decrease and points where the graph changes direction.

Second Derivative

The second derivative, \(f''(x)\), is a handy tool for understanding the behavior of a function's graph. Simply put, it's the derivative of the derivative, telling us how the rate of change, or slope, is changing over time. But what does this mean in practical terms?

• If \(f''(x) > 0\), the original function is accelerating in its increasing rate, indicating concave up.
• If \(f''(x) < 0\), the function is slowing in its increasing rate, showing concave down.
• When \(f''(x) = 0\) at a point, it suggests a potential inflection point.

At an inflection point, the graph shifts its concavity, which is why the second derivative plays a crucial role. It helps to determine these shifts and verify if the point is indeed an inflection point by checking the concavity change before and after the point.

Continuity

Continuity is all about smoothness in a function. A continuous function doesn't have breaks, jumps, or abrupt changes. If you can draw a graph without lifting your pencil, the function is continuous in that segment.
But why is continuity significant? It's crucial because it ensures that the behavior of the function is predictable. Particularly, when dealing with second derivatives and inflection points, continuity guarantees that any changes in concavity follow logically without sudden jumps.
An inflection point's presence often relies heavily on the continuity of the second derivative \(f''(x)\). For example:

• If \(f''(x)\) is continuous and changes sign at a point \(c\), then \(f''(c)\) must be zero for the curve to transition smoothly from concave up to down or vice versa.

Without continuity, this seamless transition wouldn't be possible or predictable, which would create significant difficulties in understanding and analyzing functions accurately.