Problem 78 - Thomas Calculus

Question

Can anything be said about the graph of a function \(y=f(x)\) that has a continuous second derivative that is never zero? Give reasons for your answer.

Step-by-Step Solution

Answer

The graph is consistently concave up (\(f''(x) > 0\)) or down (\(f''(x) < 0\)); no inflection points exist.

Step 1: Understand the Second Derivative

A function's second derivative, often denoted as \( f''(x) \), provides information about the concavity of the function. If \( f''(x) > 0 \) for all \( x \), the function is concave up (smiling) everywhere. If \( f''(x) < 0 \) for all \( x \), the function is concave down (frowning) everywhere.

Step 2: Continuous Second Derivative

Given that the function \( f(x) \) has a continuous second derivative that is never zero, it implies that \( f''(x) \) is either always positive or always negative across its entire domain.

Step 3: Determine Concavity and Inflection Points

Because \( f''(x) \) is never zero, there are no points where the concavity changes. Thus, there are no inflection points. The graph is either always concave up or always concave down throughout its domain.

Step 4: Conclude Graph Behavior

Since the second derivative is continuous and never zero, the function does not change its curvature, meaning it either consistently curves upwards or downwards. As a result, the graph can be seen as having a simple curvature without any twisting or flattening.

Key Concepts

Concavity

Concavity is an essential concept that helps us understand how a function's graph behaves. When we talk about concavity, we're referring to the way the graph curls or bends at different portions. Here, the second derivative, denoted as \( f''(x) \), is the hero that tells this story.

• If \( f''(x) > 0 \), the function is concave up. This means the graph looks like a smile :) and it bows upwards.
• If \( f''(x) < 0 \), the graph is concave down, resembling a frown :( as it curves downwards.

When a function’s second derivative is continuous and never zero over all \( x \), it means the entire graph maintains the same type of concavity. Therefore, it does not switch between smiling and frowning, keeping a consistent curve throughout its domain.

Inflection Points

Inflection points are interesting spots on the graph of a function where the concavity changes. This is where a graph transitions from concave up to concave down or vice versa. To find these points, we look for locations where the second derivative equals zero or is undefined.
In our scenario, where the second derivative is continuous and never zero, no such transitions occur. Consequently, there are no inflection points. Without inflection points, we can conclude that the graph has no sections that flip in its curvature. It remains either completely concave up or entirely concave down without interruption.

Graph Behavior

The behavior of a graph is closely linked to its concavity and the presence of inflection points. Understanding graph behavior allows us to visualize what the function's plot looks like over its entire domain. Since the second derivative in our function is both continuous and non-zero, the graph exhibits a uniform behavior.

• With \( f''(x) \) consistently positive, the graph always lifts upwards, like a continuous slope rising infinitely.
• If \( f''(x) \) is negative throughout, it perpetually slopes downward, like a hill curving down.

This means you can envision the graph having a smooth, uninterrupted curvature without any transitions or flat spots. It offers a simple yet predictable plot, void of twists, turns, or flattening regions.