Question

Suppose \(f^{\prime \prime}\) exists and is positive on an interval \(I\). Describe the relationship between the graph of \(f\) and its tangent lines on the interval \(I\)

Short answer

Answer: On interval \(I\), the tangent lines always lie below the graph of the function.

Step-by-step solution

Understanding the second derivative

The second derivative, \(f^{\prime \prime}\), tells us about the concavity of the function \(f\). If the second derivative is positive, it means that the function is concave up (i.e., it opens upwards like a smile). If the second derivative is negative, it means that the function is concave down (i.e., it opens downward like a frown). Since we are given that \(f^{\prime \prime}\) is positive on interval \(I\), it means that the function \(f\) is concave up on this interval.

Connecting concavity to tangent lines

When a function is concave up, it means that the function is always increasing its slope. This causes the tangent lines of the function to always lie below the graph of the function on the interval \(I\). On the other hand, if the function were concave down, the tangent lines would lie above the graph of the function.

Describing the relationship between the graph of \(f\) and its tangent lines on interval \(I\)

Since \(f^{\prime \prime}\) is positive on interval \(I\), the function \(f\) is concave up on this interval. This means that the graph of \(f\) always lies above its tangent lines on the interval \(I\). In conclusion, for the given function \(f\) with positive second derivative on interval \(I\), the tangent lines will always lie below the graph of the function on this interval.

Key concepts

Second Derivative

In calculus, the second derivative of a function, denoted as \( f''(x) \), provides insightful information about the behavior of the function's graph. It is essentially the derivative of the first derivative \( f'(x) \), meaning it measures how \( f'(x) \) changes. This change in the rate can either be an increase or decrease in the slope of the tangent line to the graph of \( f(x) \).

If the second derivative \( f''(x) \) is positive, it indicates that the function's slope \( f'(x) \) is increasing, causing the graph to curve upward. In this scenario, the function is said to be **concave up**. Conversely, if \( f''(x) \) is negative, the slope \( f'(x) \) is decreasing, leading the graph to curve downward, and so the function is **concave down**. In situations where \( f''(x) = 0 \), the graph may have an inflection point where the concavity changes value.

Understanding the second derivative is crucial as it allows you to predict the shape and curvature of the function's graph, offering a deeper comprehension of its graphical behavior.

Concavity

Concavity defines the direction in which a function's graph curves. It helps to identify whether the graph is opening upward or downward. When a function is **concave up**, it has a shape similar to an open bowl or a smile, indicating that the graph's slope is progressively increasing.

Key characteristics include:

• The second derivative \( f''(x) \) is positive.
• The function's slopes become steeper as you move from left to right.
• Any tangent drawn to the function's curve lies below the graph.

Conversely, a **concave down** function resembles an upside-down bowl or a frown, where the graph's slope is decreasing:

• The second derivative \( f''(x) \) is negative.
• The function's slopes become less steep from left to right.
• Tangents drawn to the curve lie above the graph.

The concept of concavity is integral in understanding the nature of graphs, particularly while sketching them or analyzing critical points.

Tangent Lines

A tangent line is a straight line that touches a function's curve at exactly one point and moves in the same direction as the curve at that point. The slope of this line represents the derivative of the function at that particular point.

In a concave up scenario, as the function's graph is curving upward, the tangent line at any point will be below the curve because the slope is increasing. This visually illustrates how the slope of the function is getting steeper as you move along the interval.

Conversely, in a concave down scenario, the tangent lines will lie above the curve, showing that the slope of the function is decreasing. Drawing tangent lines can not only help in visualizing the behavior of the graph at individual points but also in understanding how the function as a whole is changing over certain intervals.

Function Graphs

A function graph is a visual representation of the relationship between variables in a function. It shows how the value of one variable depends on another and provides valuable insight into the function's behavior.

When plotting a function's graph, understanding how derivatives work is crucial as they dictate how the graph is shaped and curved. The first derivative \( f'(x) \) gives the slope of the tangent line, indicating whether the function is increasing or decreasing at any given point.

The second derivative \( f''(x) \) shapes the curvature, or concavity, as we've seen above, showcasing whether the graph opens up or down. These characteristics help in understanding features such as:

• Change in direction and speed of function values.
• Turning points, such as local maxima and minima.
• Points of inflection where concavity changes.

Analyzing function graphs using derivatives makes it easier to predict future values and comprehend the broader picture of the function's behavior.