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Chapter 3: Problem 72

Problems \(66-72\) go back to the theory. Create a function \(y(x)\) that has its maximum at a rough point and its minimum at an endpoint.

Short Answer

Expert verified
Use \( y(x) = 1 - |x| \) on \([-1, 1]\) to have maximum at cusp and minimum at endpoints.

Step by step solution

01

Identify the Requirements

We need to create a function \( y(x) \) such that it has a maximum at an internal point (not smooth, possibly a cusp or discontinuity) and a minimum at an endpoint of the interval. A common function type that has a cusp is the absolute value function, and it might be useful here.
02

Choose a Suitable Function

Consider the simple absolute value function \( y(x) = |x| \), which has a cusp at the origin and is continuous. This function has a sharp point at \( x = 0 \).
03

Define the Interval for the Function

Since the minimum is required to be at an endpoint, we decide on an interval such as \([-1, 1]\). This will make it easier to identify the endpoints and apply any modifications.
04

Verify Maximum and Minimum Points

For the function \( y(x) = |x| \) on \([-1, 1]\): - The minimum occurs at the endpoints, \( x = -1 \) and \( x = 1 \), where \( y(x) = 1 \).- The function has a cusp at \( x = 0 \), which becomes the maximum point at \( y(0) = 0 \).
05

Adjust and Ensure Conditions Are Met

Consider a modified form \( y(x) = 1 - |x| \) on \([-1, 1]\) to swap conditions,- The maximum is now at the cusp, \( x = 0 \) where the value is \( y(0) = 1 \).- The minimum occurs at \( x = -1 \) and \( x = 1 \) with \( y(x) = 0 \).This function meets the problem requirements.

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Key Concepts

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

Absolute Value Function
An absolute value function is a fundamental mathematical concept represented by the notation \( y(x) = |x| \). The absolute value of a number is its distance from zero on the number line, without considering direction. Thus, it is always non-negative.
  • Graphical Representation: The graph of \( y(x) = |x| \) is V-shaped and has a vertex at the origin (0,0), forming a sharp point known as a cusp. This trait makes the absolute value function unique as it is continuous but not differentiable at this point.
  • Key Properties: The function has two linear segments. For \( x > 0 \), it behaves like \( y(x) = x \), while for \( x < 0 \), it acts as \( y(x) = -x \). Therefore, \( y(x) \) transitions abruptly at \( x = 0 \). This discontinuity in the slope is why it is a popular choice when designing functions with special maximum or minimum conditions.
Its characteristic sharp point allows it to be utilized effectively in problems requiring non-smoothness or cusps.
Maximum and Minimum Points
Finding the maximum and minimum points of a function involves determining where the function reaches its highest or lowest values within a given domain.
  • Maximum Point: This occurs where the value of the function is higher compared to its surroundings. In the context of the absolute value function \( y(x) = 1 - |x| \), the peak, or cusp, at \( x = 0 \) becomes the maximum point. Here, the function evaluates to \( y(0) = 1 \), which is the highest value on the interval specified.
  • Minimum Point: The minimum values are found at the endpoints of the interval. For the specified interval \( [-1, 1] \), these points are \( x = -1 \) and \( x = 1 \) where the function evaluates to zero, making them the lowest points.
These points are significant in mathematical analysis and have various applications, such as optimizing solutions in calculus and ensuring that function properties meet problem constraints.
Piecewise Functions
Piecewise functions provide a versatile way to define functions that have different expressions based on the input value.
  • Definition: A piecewise function is written as a series of conditional equations. This allows the function to "change its rule" depending on which interval of values \( x \) is in. \( y(x) = |x| \) is inherently piecewise because it toggles between \( y(x) = x \) when \( x \) is non-negative and \( y(x) = -x \) when \( x \) is negative.
  • Use in Variable Conditions: Such functions are particularly useful in scenarios where different behavior is required at different ranges of \( x \). For example, setting a maximum at a specific point and minimum at the endpoints as seen with \( y(x) = 1 - |x| \).
This adaptability makes piecewise functions a powerful tool in calculus when solving complex problems with specific conditions tailored to different segments of a domain.

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