Problem 80 - University Calculus: Early Transcendentals

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

The function value at \(x = -c\) is the same maximum as at \(x = c\), due to symmetry.

Step 1: Understand Even Functions

An even function is symmetric about the y-axis, which means that for any point \(x\), \(f(x) = f(-x)\). This property will be crucial in solving the problem.

Step 2: Evaluate the Given Condition

We know that \(f(x)\) has a local maximum value at \(x=c\). This means that \(f(c)\) is greater than or equal to \(f(x)\) for values of \(x\) near \(c\).

Step 3: Use the Symmetry Property

Since \(f(x)\) is an even function, \(f(c) = f(-c)\). Therefore, whatever value \(f(c)\) holds, \(f(-c)\) holds the same value.

Step 4: Draw the Conclusion

Because \(f(c) = f(-c)\), and \(f(c)\) is a local maximum, \(f(-c)\) also holds this local maximum value. Thus, \(x=-c\) will also be a point where the function reaches this same maximum value due to the symmetry of even functions.

Key Concepts

Symmetric Property

The symmetric property is an essential feature when dealing with even functions. This means that if you have a function, say \(f(x)\), classified as even, it will exhibit symmetry about the y-axis. This property is mathematically expressed as \(f(x) = f(-x)\), signifying that each point on the graph of the function across the y-axis is mirrored.

This concept is important, for instance, when evaluating points \(x=c\) and \(x=-c\). Due to symmetry, the value of \(f(c)\) must be the same as that of \(f(-c)\).

In practical terms, imagine folding a piece of paper along the y-axis wherein the graph of the function is plotted. If the curve aligns perfectly with itself, the function indeed reflects symmetry.

• Helps to quickly determine function values.
• Supports understanding of even functions better.
• Makes graphing these functions simpler.

In problems like the one presented, not only does this simplify calculations significantly, but it also readily provides insights that might not be apparent at first glance.

Local Maximum

A local maximum in a function's graph occurs at a point where the function value is greater than those in its immediate vicinity. This tells us that among values close to this point, it is the peak.

For example, when \(f(x)\) is an even function and reaches a local maximum at \(x=c\), it's mathematically denoted as: \(f(c) \geq f(x)\) for values of \(x\) around \(c\). This essentially describes a 'hill' in the graph.

• Indicates 'high points' for a given region of the function.
• Useful to identify optimal values in real-life scenarios.

If you know a function has symmetry, as in the case of even functions, you can infer more from local maximum points. At these points—and their symmetric counterparts—the function value will remain constant due to their mirrored nature. This insight can be pivotal for both analysis and practical applications, such as optimization problems.

Y-Axis Symmetry

Y-axis symmetry is a specific type of symmetry showcased by even functions. If a function's graph is symmetric with respect to the y-axis, any change in \(x\) will not affect the function value since \(f(x) = f(-x)\).

This type of symmetry suggests that the function does not favor one side of the y-axis, allowing balanced computations when dealing with opposite x-values. For instance, if a maximum occurs at \(x=c\), exactly the same maximum will reflect at \(x=-c\).

• Simplifies the graphing of even functions.
• Ensures the function's behavior is predictable across the y-axis.
• Aids in understanding the function's overall symmetry.

Understanding y-axis symmetry is a vital skill in mathematics, as it permits one to draw conclusions about the function at mirrored points easily. This skill is particularly beneficial when trying to find function values or insights about its behavior over different domains, as seen in the provided exercise.