Problem 64 - Thomas Calculus

Question

Even functions If an even function \(f(x)\) has a local maximum value at \(x=c,\) can anything be said about the value of \(f\) at \(x=-c ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

Yes, f(-c) = f(c) since f is even, so there's also a local max at x = -c.

Step 1: Define Even Functions

An even function is defined as a function where \( f(-x) = f(x) \) for every \( x \) in its domain.

Step 2: State the Given Condition

The function \( f(x) \) has a local maximum at \( x = c \). This means that there is an interval \((a, b)\) around \( c \) such that \( f(c) \geq f(x) \) for all \( x \) in \((a, b)\).

Step 3: Apply Even Function Property to c

Since \( f(x) \) is an even function, \( f(-c) = f(c) \). Therefore, the value of the function at \( x = -c \) is the same as at \( x = c \).

Step 4: Conclude about Value at -c

As \( f(c) \) is a local maximum and \( f(-c) = f(c) \), the value of \( f(x) \) at \( x = -c \) must also be \( f(c) \). Hence, if \( f(x) \) has a local maximum at \( x = c \), it will also have a local maximum at \( x = -c \).

Key Concepts

Local Maximum

A local maximum of a function occurs at a point where the function's value is greater than or equal to all nearby values. Imagine climbing a hill, and you pause at the top of the hill; that spot is a local maximum.
For a more mathematical perspective, suppose you have a function \( f(x) \) and it is said to have a local maximum at \( x = c \) if there exists an interval \( (a, b) \) such that the following condition holds:

• \( f(c) \geq f(x) \) for all \( x \) in the interval \( (a, b) \).

This means around that point \( c \), the function does not get any larger.
In the context of even functions, this special property has implications on symmetry, which means if \( x = c \) is a local maximum, then \( x = -c \) also shares the same local maximum value.

Symmetry in Functions

The concept of symmetry in functions simplifies understanding of the behavior of such functions across their domain. When a function is even, it exhibits symmetry about the y-axis. This means you can fold the graph along this axis, and both sides would match perfectly.
This symmetry is mathematically represented as:

• Even functions satisfy \( f(-x) = f(x) \) for any value \( x \).

This means that the function's shape, and thus its local maxima and minima, will appear mirrored on either side of the y-axis. For example, if you know that an even function achieves its highest value at some point, say \( x = c \), you immediately know it also reaches this value at \( x = -c \) due to symmetry. This understanding helps predict behavior across the entire function, simplifying graphing and analysis.

Properties of Functions

Functions hold various properties that define their behavior and categorization, with evenness being one of these characteristics. Properties like continuity, differentiability, and symmetry are crucial in understanding how functions behave.

• Continuity: Even functions are often continuous, meaning they have no abrupt jumps or breaks in their graph.
• Symmetry: As discussed, even functions reflect symmetrical behavior about the y-axis.
• Differentiability: While not all even functions are differentiable, many are, allowing for more in-depth analysis through calculus.

These properties can greatly influence how functions are graphed and analyzed. When grappling with exercises involving even functions, considering these fundamental attributes helps demystify why certain points are local maxima or minima, and how changes reflect across the function's graph.