Problem 19 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. If a function \(f\) has an absolute minimum on \((a, b),\) then there is a critical point of \(f\) in \((a, b) .\)

Step-by-Step Solution

Answer

False, the statement is not always true.

Step 1: Understand the Problem

The problem asks us to determine if a function \( f \) having an absolute minimum on the open interval \( (a, b) \) ensures the existence of a critical point of \( f \) within this interval. A critical point occurs when the derivative is zero or undefined.

Step 2: Review Definitions

Recall that a function \( f \) has a critical point at \( c \) within \( (a, b) \) if \( f'(c) = 0 \) or \( f' \) is undefined at \( c \). An absolute minimum at \( c \) means \( f(c) \leq f(x) \) for all \( x \) in the domain.

Step 3: Consider Counterexamples

Consider the function \( f(x) = x^3 \). It doesn't have a critical point on \( (-1, 1) \) as its derivative \( f'(x) = 3x^2 \) is never zero or undefined. However, \( f(x) \) achieves its absolute minimum value of 0 at \( x = 0 \) within interval \((-1, 1)\).

Step 4: Conclude from Example

The example shows that a function can have an absolute minimum without having a critical point within the interval. Hence, the statement that a function must have a critical point if it has an absolute minimum in \((a, b)\) is false.

Key Concepts

Absolute Minimum

When we talk about an absolute minimum of a function, we are referring to the lowest point that the function achieves. This means that at this point, the value of the function is less than or equal to the value at any other point in the domain of interest. For a function \( f \) defined on an interval \( (a, b) \), if there is a point \( c \) in that interval such that \( f(c) \leq f(x) \) for all \( x \) in \((a, b)\), then \( c \) is an absolute minimum point.
When looking for an absolute minimum, it is important to verify that all values of the function elsewhere in the interval don't drop below the value at this point. This concept is essential in various applications, from optimizing costs to finding the lowest points in physical systems. The idea is quite simple but powerful, as it helps in identifying key parts of a function's behavior. Absolute minimization can also be significantly impacted by boundary conditions, which aren't part of open intervals.

Derivative

The derivative of a function gives us an insight into the function's rate of change at any given point. Mathematically, the derivative of a function \( f \) at a point \( x \) is denoted as \( f'(x) \). It tells us how the function is "moving" at \( x \) — whether it is increasing, decreasing, or remaining constant.

• If \( f'(x) > 0 \), the function is increasing at \( x \).
• If \( f'(x) < 0 \), the function is decreasing at \( x \).
• If \( f'(x) = 0 \), the function has a horizontal tangent or is flat at that point. This could be a potential critical point.

The concept of the derivative is pivotal for finding critical points, where the function's slope is zero or undefined. These are candidate locations for possible maximum or minimum values of the function. Understanding how to calculate and interpret derivatives is a core component of calculus, crucial for solving real-world problems involving changing systems.

Open Interval

An open interval, seen as \( (a, b) \), refers to all the points that lie between \( a \) and \( b \) without including the endpoints themselves. This means we are not considering the values \( a \) and \( b \) as part of the interval. Hence, points closer or equal to these endpoints are excluded.
Open intervals are particularly useful when examining the behavior of functions within a specified range without worrying about boundary conditions. It comes to play significantly in calculus when determining the behavior of functions related to concepts like continuity, limits, and differentiation. For example, discussing the presence of minima and maxima explicitly requires understanding intervals to know which values are reasonable to consider within the problem's context.

• Open intervals focus on the inside values strictly.
• They are crucial in theoretical problems where an endpoint's exact value does not influence the analysis.

Counterexample

A counterexample is a specific case or example that refutes a statement or hypothesis. In mathematical problems, finding a counterexample is a powerful method to demonstrate that a general claim or assumption does not hold universally.
For instance, in the given problem, we consider the function \( f(x) = x^3 \) on the interval \((-1, 1)\). While this function achieves an absolute minimum value at \( x = 0 \), the derivative \( f'(x) = 3x^2 \) is never zero or undefined in the interval. This highlights that, although the function has a minimum, there is no associated critical point, thus providing a counterexample to disprove the initial statement. Counterexamples provide:

• Solid proof to show limitations of theorems or statements.
• Insightful understanding into the precise conditions needed for statements to be true.
• Clarification of cases where additional conditions may be required.

Overall, recognizing and utilizing counterexamples is a cornerstone of mathematical reasoning and proof.