Question

A function \(f(x)\) is given. (a) Find the possible points of inflection of \(f\). (b) Create a number line to determine the intervals on which \(f\) is concave up or concave down. \(f(x)=x^{2}-2 x+1\)

Short answer

(a) No inflection points. (b) Concave up on \((-\infty, +\infty)\).

Step-by-step solution

Find the Second Derivative

A point of inflection occurs where the second derivative of the function changes sign. Start by finding the first derivative of the function \( f(x) = x^2 - 2x + 1 \), which is \( f'(x) = 2x - 2 \). Then, find the second derivative: \( f''(x) = 2 \).

Set the Second Derivative to Zero

Set the second derivative equal to zero to find potential inflection points: \( f''(x) = 2 = 0 \). Since this equation has no solutions, there are no potential points of inflection for the function.

Analyze the Sign of the Second Derivative

Since \( f''(x) = 2 > 0 \) for all \( x \), \( f \) is concave up on its entire domain (\( -\infty, +\infty \)).

Summarize Concavity Intervals

Based on the second derivative, indicate that \( f(x) \) is concave up on the interval \( (-\infty, +\infty) \) and there are no intervals where it is concave down.

Key concepts

Concavity

Concavity is a fundamental concept in calculus which helps us understand the curvature direction of a function. Imagine pressing down on the middle of a flexible ruler; if it bends upwards like a smile, the function is concave up. If it bends downwards like a frown, the function is concave down.

To determine the concavity of a function, we look at its second derivative:

• If the second derivative is positive (\( f''(x) > 0 \)), the function is concave up.
• If the second derivative is negative (\( f''(x) < 0 \)), the function is concave down.

In our exercise, since the second derivative \( f''(x) = 2 \) is always positive, the function is concave up over its entire domain from \(-\infty \text{ to } +\infty\). This indicates that our function's graph always curves upwards.

Points of Inflection

Points of inflection are special points on the graph of a function where the concavity changes; for example, from concave up to concave down, or vice versa. At these points, the second derivative of the function changes sign.

However, not every point where the second derivative is zero is a point of inflection. It also must result in a change in the sign of the second derivative:

• To identify potential points of inflection, solve \( f''(x) = 0 \) and find where it changes signs.
• If no sign change occurs, no inflection point exists.

In our example, solving \( f''(x) = 2 \) gives us no solutions, indicating the function has no points of inflection.

Second Derivative

The second derivative of a function, denoted as \( f''(x) \), provides valuable information about the function's concavity and potential points of inflection. It represents the derivative of the first derivative, or the rate at which the slope of the function is changing.

Here’s a step-by-step approach to find and use the second derivative:

• First, differentiate the function to get the first derivative \( f'(x) \).
• Then, differentiate again to find the second derivative \( f''(x) \).
• Analyze \( f''(x) \) to understand the function's concavity and locate points of inflection.

In our example, the initial function \( f(x) = x^2 - 2x + 1 \) turned into \( f'(x) = 2x - 2 \), and then \( f''(x) = 2 \), which played a crucial role in analyzing concavity.

Function Analysis

Function analysis is a comprehensive process used to understand how a function behaves across its domain. By analyzing derivatives, particularly the second derivative, we can determine characteristics such as concavity, extrema, and points of inflection.

Function analysis typically involves these steps:

• Calculate the first derivative to find critical points where the function's slope is zero.
• Use the second derivative to determine concavity and locate potential points of inflection.
• Construct number lines for visualization, indicating where the function is concave up or down.
• Summarize the complete behavior of the function based on these observations.

In our problem, knowing \( f''(x) = 2 \) helped us comprehensively analyze the function \( f(x) \) as being concave up on \(-\infty, +\infty \), with no points of inflection, providing a complete picture of its behavior.