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Chapter 4: Problem 84

Given that \(f^{\prime}(x)\) is continuous everywhere and changes from negative to positive at \(x=a,\) which of the following statements must be true? (a) \(a\) is a critical point of \(f(x)\) (b) \(f(a)\) is a local maximum (c) \(f(a)\) is a local minimum (d) \(f^{\prime}(a)\) is a local maximum

Short Answer

Expert verified
Statements (a) and (c) are true.

Step by step solution

01

Understand Critical Points

A critical point occurs where \(f'(x) = 0\) or \(f'(x)\) is undefined. Given \(f'(x)\) changes from negative to positive at \(x=a\), it implies \(f'(a) = 0\). Thus, \(a\) is a critical point of \(f(x)\).
02

Analyze the Behavior of f'(x)

Since \(f'(x)\) changes from negative to positive at \(x=a\), it indicates that the slope of \(f(x)\) changes from decreasing to increasing. This often signifies that \(f(x)\) has a local minimum at \(x=a\).
03

Assess if f(a) is a Local Maximum

A local maximum occurs when \(f'(x)\) changes from positive to negative, not negative to positive. Thus, \(f(a)\) cannot be a local maximum.
04

Consider f'(a) as a Local Maximum

For \(f'(a)\) to be a local maximum, its adjacent values would need to be lower than \(f'(a)\). However, since \(f'(x)\) changes from negative to positive, it transitions through \(f'(a) = 0\), not through a peak; this doesn鈥檛 characterize a local maximum at \(f'(a)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus
Calculus is a branch of mathematics that focuses on rates of change and the accumulation of quantities. It provides the tools needed to understand the behavior of functions and their derivatives.
When working with functions, one major application of calculus is identifying critical points and understanding their significance. By examining the derivative of a function, we can assess the increasing or decreasing trends and identify specific points of interest such as local maxima or minima.
  • The derivative of a function, denoted as \( f'(x) \), gives us the slope of the tangent to the function at any given point.
  • If \( f'(x) = 0 \), it indicates that the slope is horizontal, suggesting a potential critical point.
  • If \( f'(x) \) is undefined, it can also indicate a critical point, depending on the context of the function.
Understanding these basic concepts is essential for analyzing the behavior of functions through calculus.
Local Minimum
In calculus, a local minimum refers to a point where a function's value is lower than any nearby points. Specifically, it's a spot that represents a 'valley' on the graph of the function.
For a function \( f(x) \), if the derivative \( f'(x) \) changes from negative to positive at a point \( x = a \), this behavior typically indicates a local minimum. This is depicted visually by the curve of \( f(x) \) transitioning from descending to ascending slope.
  • When approaching a local minimum from the left, the function decreases.
  • After passing through the minimum, as you move to the right, the function begins to increase.
  • This change in direction of the slope from negative to positive is what characterizes a local minimum.
Derivative Analysis
Derivative analysis involves examining the behavior of a function by analyzing its derivatives. This process is crucial for finding critical points and understanding the shape and movement of the function.
When performing derivative analysis, understanding how the derivative \( f'(x) \) behaves is key. A derivative crossing through zero is an essential feature in identifying critical points.
  • If \( f'(x) \) transitions from negative to positive at \( x = a \), then the function attains a local minimum at \( x = a \).
  • Conversely, if \( f'(x) \) changes from positive to negative, the point \( x = a \) is a local maximum.
  • If neither sign change occurs, \( x = a \) may be an inflection point but not necessarily a critical one.
Derivative analysis helps us comprehensively understand how a function's graph behaves around specific points, assisting in determining whether these points are peaks, troughs, or points of inflection.

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