Problem 48 - Calculus

Question

True or false? Give an explanation for your answer. If \(f^{\prime \prime}(x)>0\) then \(f^{\prime}(x)\) is increasing.

Step-by-Step Solution

Answer

True, if \( f''(x) > 0 \), then \( f'(x) \) is increasing.

Step 1: Understand the meaning of f''(x)>0

The condition \( f''(x) > 0 \) means that the second derivative of the function \( f(x) \) is positive. In calculus, if the second derivative is positive, it indicates that the function \( f(x) \) is concave up.

Step 2: Understand the relation between f'(x) and f''(x)

The second derivative \( f''(x) \) represents the rate of change of the first derivative \( f'(x) \). Since \( f''(x) > 0 \), it implies that \( f'(x) \) is increasing. This is because a positive second derivative means the slope of the original function is getting steeper or more positive.

Step 3: Conclude with true or false

Given the understanding from previous steps, if \( f''(x) > 0 \) then \( f'(x) \) is indeed increasing. Therefore, the given statement is true based on the relationship between the first and second derivatives.

Key Concepts

Second Derivative

The second derivative of a function, denoted as \( f''(x) \), provides insightful information about the behavior of the original function \( f(x) \). When we talk about the second derivative, we are essentially considering the rate at which the first derivative, \( f'(x) \), changes. In simpler terms, the second derivative tells us how the slope of the tangent line to \( f(x) \) is evolving as we move along the curve.

When \( f''(x) > 0 \), it indicates that the curve of \( f(x) \) is concave up, which looks like a smile. Concave up means that the function is bending upwards. Think of it like the shape of a bowl or a u-shape.

• If the graph of a function is concave up, it implies the slopes of tangent lines are becoming more positive or less negative.
• This curvature suggests that \( f'(x) \) is increasing over the interval.

Understanding the second derivative is crucial as it helps in analyzing the acceleration and curvature of the function, revealing a deeper insight into function behavior beyond just the increasing or decreasing nature of \( f(x) \).

Increasing Function

An increasing function is one that grows larger or stays the same as the input value \( x \) increases. It's important to note that for a function \( f(x) \) to be considered increasing, its slope, given by the first derivative \( f'(x) \), must be positive or zero over the relevant interval.

When \( f''(x) > 0 \), indicating a concave-up shape, the first derivative \( f'(x) \) increases, which means the slope is becoming steeper at every point. As the slope keeps getting steeper, \( f(x) \) is definitely moving upward faster.

• Although \( f(x) \) itself doesn't have to be increasing everywhere, it indicates a tendency towards moving upwards over the specific interval where \( f''(x) > 0 \).
• Understanding whether a function is increasing helps predict its overall trend and pattern as \( x \) changes.

Recognizing when a function is increasing or decreasing is a powerful insight when plotting graphs and understanding real-life scenarios modeled by functions.

First Derivative

The first derivative, expressed as \( f'(x) \), is a fundamental tool in calculus detailing the instantaneous rate of change or slope of a function \( f(x) \). Essentially, it tells us how quickly \( f(x) \) changes at each point along the curve.

By observing the first derivative, one can determine whether the function is increasing or decreasing. Specifically:

• When \( f'(x) > 0 \), \( f(x) \) is increasing, showing that the function's output grows as \( x \) increases.
• When \( f'(x) < 0 \), \( f(x) \) is decreasing, meaning the function’s output falls as \( x \) grows.

The relationship between the first and second derivatives is key in determining concavity and the nature of \( f(x) \). Given \( f''(x) > 0 \), the first derivative \( f'(x) \) will be increasing. This increasing behavior of \( f'(x) \) implies that the rate of change itself is getting more intense or progressively larger, showcasing a steeper ascent in \( f(x) \)'s curve.

Understanding the first derivative heavily aids in graphing functions accurately and assessing their dynamics over different intervals.