Problem 14 - Calculus Early Transcendentals

Question

Assume that f is differentiable everywhere. Determine whether the statement is true or false. Explain your answer. If \(f^{\prime}\) is increasing on \([0,1]\) and \(f^{\prime}\) is decreasing on \([1,2],\) then \(f\) has an inflection point at \(x=1\)

Step-by-Step Solution

Answer

The statement is true; \( f \) has an inflection point at \( x = 1 \).

Step 1: Understand the Conditions

We are given that the function \( f(x) \) is differentiable everywhere. Additionally, \( f'(x) \), which is the derivative of \( f(x) \), is increasing on \([0, 1]\) and decreasing on \([1, 2]\). We need to determine if \( f \) has an inflection point at \( x = 1 \).

Step 2: Recall Definition of Inflection Point

An inflection point occurs at \( x = a \) if the second derivative \( f''(x) \) changes sign at \( x = a \). This means \( f''(x) \) must exist around \( x = a \) and should switch from positive to negative, or vice versa.

Step 3: Analyze f'(x) Behavior

Given that \( f'(x) \) is increasing on \([0, 1]\) and decreasing on \([1, 2]\), this indicates that \( f''(x) > 0 \) on \([0, 1)\) and \( f''(x) < 0 \) on \((1, 2]\).

Step 4: Determine the Sign Change at x=1

At \( x = 1 \), \( f''(x) \) changes from positive on the left (\([0,1)\)) to negative on the right (\((1,2]\)). This means \( f''(x) \) changes sign at \( x = 1 \).

Step 5: Conclusion on Inflection Point

Since the second derivative \( f''(x) \) changes sign at \( x = 1 \), \( f \) has an inflection point at this location. Hence, the statement is true.

Key Concepts

Differentiability

Differentiability is a fundamental concept in calculus that deals with whether a function has a derivative at every point in its domain. When we say a function is differentiable everywhere, it means you can find its derivative at every single point across its domain. In other words, the function is smooth and doesn't have corners, cusps, or any sort of abrupt changes.

• If a function is differentiable at a point, it is also continuous at that point, but the converse is not necessarily true.
• Differentiability implies the existence of a tangent line at every point.
• Smooth curves and surfaces are differentiable, while sharp edges are not.

Understanding differentiability is crucial because it opens the door to using calculus tools like derivatives and integrals effectively. In the given exercise, knowing that the function is differentiable everywhere allows us to confidently work with its derivatives, leading us to further insights about its behavior, especially regarding inflection points.

Inflection Point

An inflection point is a point on the graph of a function where the curvature changes direction. It is a critical concept when analyzing the concavity of a function.

• An inflection point occurs when the second derivative changes sign, indicating a change in concavity.
• If the function transitions from being concave up to concave down or vice versa, it will have an inflection point.
• Inflection points are important for determining the shape and behavior of a graph.

In the original exercise, it is given that the derivative of the function first increases and then decreases. This behavior suggests that the second derivative changes from positive to negative as x crosses a certain point, namely \( x = 1 \), indicating an inflection point.

Second Derivative

The second derivative of a function, often represented as \( f''(x) \), provides information about the concavity of the function. Concavity describes whether a function is curving upwards or downwards at a particular interval.

• When \( f''(x) > 0 \), the function is concave up, resembling a U-shape.
• When \( f''(x) < 0 \), the function is concave down, resembling an n-shape.
• The points where \( f''(x) = 0 \) are potential candidates for inflection points, provided there's a sign change.

In the provided solution, the analysis of the second derivative is crucial because it allows us to confirm the presence of an inflection point at \( x = 1 \). By showing that \( f''(x) \) changes sign here, we understand the function transitions from one concavity to another.

Sign Change

Sign change is a significant event in calculus, especially when looking at derivatives. It indicates a transition in the behavior of a function.

• A sign change in the first derivative \( f'(x) \) signals a local maximum or minimum, showing where the function's direction changes.
• A sign change in the second derivative \( f''(x) \) signals an inflection point, indicating a change in concavity.
• Tracking these changes helps in understanding not just where these critical points occur, but also how the function behaves around them.

In the context of the original exercise, the concept of a sign change applies to the second derivative around \( x = 1 \). This change from positive to negative ensures that there indeed is a shift in the concavity of the function, validating the presence of an inflection point at that x-value.