Question

Multiple Choice If an even function \(f\) with domain all real numbers has a local maximum at \(x=a,\) then \(f(-a)\) (A) is a local minimum. (B) is a local maximum. (C) is both a local minimum and a local maximum. (D) could be either a local minimum or a local maximum. (E) is neither a local minimum nor a local maximum.

Short answer

(B) is a local maximum.

Step-by-step solution

Understand The Property Of an Even Function

One important property of even functions is their symmetry about the y-axis, which can be written mathematically as \(f(-x) = f(x)\) for all \(x\) in the function's domain.

Apply The Even Function Property to the Problem

Given that there is a local maximum at \(x=a\), we can use the even function property to reason that there will also be a local maximum at \(x=-a\), since \(f(-x) = f(x)\).

Identify the Correct Answer Choice

Thus, \(f(-a)\) must also be a local maximum, which corresponds to choice (B).

Key concepts

Local Maximum

When we discuss the concept of a local maximum in the context of a function, such as an even function, we are referring to a point on the function's graph where its y-value is greater than those of the points immediately surrounding it. Imagine standing on the top of a small hill; at the peak, you are at the local maximum height relative to the ground directly near you. Similarly, in a mathematical function, when you're at the local maximum, the function's value doesn't get higher for those nearby x-values.

Now, just as hills have their unique shapes and sizes, local maximums can vary widely in their appearance on a graph. However, one thing remains constant: they represent a peak in the function's value for that small region. For an even function with a local maximum at some point, say \( x=a \), this peak allows us to make specific predictions about the rest of the graph due to the function's inherent symmetry, which we'll explore shortly in more detail.

Symmetry About the Y-Axis

Symmetry about the y-axis is a distinct characteristic of even functions. This symmetry can be visually understood by picturing a mirror placed along the y-axis of a graph; the part of the function's graph on one side of the y-axis is the mirror image of the other side. Mathematically, this symmetry implies that for every point \( (x, f(x)) \) on the graph, there is a corresponding point \( (-x, f(x)) \) that has the same y-value. The formal definition of this is \( f(-x) = f(x) \) for all \( x \) within the function's domain.

But why does this matter for solving our exercise? When an even function has a local maximum at \( x=a \), due to this mirror-like property, there must also be a local maximum at \( x=-a \). The heights of these peaks (y-values) are identical, even though they exist at different places along the x-axis. This concept is critical because it helps us to recognize that local maximums and minimums have partners on the opposite side of the y-axis when dealing with even functions.

Function Domain

A function's domain is the set of all possible input values (commonly represented as x) for which the function is defined. It’s like having a specific set of keys and determining which ones fit into a particular lock. In the realm of mathematics, not every x-value can work with every function — some just don’t “fit.” The domain includes all the x-values that do, and one must consider it with careful attention in any mathematical problem involving functions.

For even functions, it is often the case, as it is with our exercise, that the domain includes all real numbers because the nature of their symmetry allows for a value at every point along the x-axis. However, not all functions are so inclusive; some have restrictions due to the presence of certain mathematical operations, like division by zero or square roots of negative numbers.

Determining a function's domain is essential when exploring properties like local maximums, as the domain informs us where we can look for such features and where to expect symmetry, if any. Understanding the domain can lead us to a deeper comprehension of the function's behavior across all possible x-values.