Problem 70 - Thomas Calculus in SI Units

Question

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, \(f(-c) = f(c)\), so \(f(-c)\) is also a local maximum.

Step 1: Understanding Even Functions

An even function satisfies the property that for every number \(x\), \(f(x) = f(-x)\). This means that the graph of the function is symmetric with respect to the \(y\)-axis.

Step 2: Analyzing the Given Problem

We are given that the function \(f(x)\) is even and has a local maximum at \(x = c\). We need to determine the behavior of \(f(x)\) at \(x = -c\).

Step 3: Applying the Even Function Property

Since \(f(x)\) is even, \(f(c) = f(-c)\) must hold true. This symmetry implies that whatever functional value \(f(c)\) attains (including a local maximum or minimum), \(f(-c)\) must have the same value.

Step 4: Concluding the Solution

As such, if \(f(c)\) is a local maximum, then \(f(-c) = f(c)\) is also a local maximum at \(x = -c\). This conclusion uses the property that the function is symmetric around the \(y\)-axis.

Key Concepts

local maxima

In calculus, local maxima are points where a function reaches a peak relative to values immediately around it. A local maximum at point \(x = c\) means that \(f(c)\) is larger than any \(f(x)\) for \(x\) in some small neighborhood around \(c\).
If we consider a function \(f(x)\) as having a local maximum at \(x = c\), we know something specific about its behavior near \(c\).

• Within the range of immediately surrounding points, no value of \(f(x)\) is higher than \(f(c)\).
• This does not necessarily mean \(f(c)\) is the highest value overall—just locally.

When dealing with even functions, this becomes even more intriguing due to symmetry. If \(f(c)\) is a local maximum and \(f(x)\) is even, then both \(f(c)\) and \(f(-c)\) attain the local maximum value due to the function's symmetry.

function symmetry

Symmetry in functions refers to a balanced and mirrored nature of their graphs. For even functions, this specifically means that the graph is a mirror image when cut along the \(y\)-axis.
Mathematically, an even function follows the rule \(f(x) = f(-x)\). Here’s what this signifies:

• The right and left sides of the graph (relative to the \(y\)-axis) look identical.
• Any y-coordinate you find at \(x = c\) can also be found at \(x = -c\).

When you grasp this, it helps clarify why if there's a local maximum at \(x = c\), there must be one at \(x = -c\) too for even functions. The symmetry about the \(y\)-axis plays a crucial part in predicting function behavior at opposite points.

calculus concepts

Calculus involves concepts that help us understand changes and values of different functions. For even functions and local maxima, these calculus concepts become vital.

• Differentiation helps identify where functions have peaks or valleys, known as extrema.
• Knowing \(f'(x) = 0\) at \(x = c\) suggests a potential local maximum or minimum.
• Further evaluating if \(f''(x) < 0\) confirms a local maximum at that point.

Given an even function \(f(x)\), once you determine \(x = c\) as a local maximum using these criteria, symmetry directly tells you've another local maximum at \(x = -c\). Exploring the role of derivatives provides the analytical method, while symmetry details the geometric reason.