Question

Even and odd functions a. Suppose a nonconstant even function \(f\) has a local minimum at C. Does \(f\) have a local maximum or minimum at \(-c ?\) Explain. (An even function satisfies \(f(-x)=f(x)\) -) b. Suppose a nonconstant odd function \(f\) has a local minimum at C. Does \(f\) have a local maximum or minimum at \(-c ?\) Explain. (An odd function satisfies \(f(-x)=-f(x) .)\)

Short answer

Answer: For an even function, if there is a local minimum at point \(c\), there will also be a local minimum at the symmetric point \(-c\). For an odd function, if there is a local minimum at point \(c\), there will be a local maximum at the symmetric point \(-c\).

Step-by-step solution

Remember the definition of even functions

An even function is a function that satisfies the equation \(f(-x) = f(x)\). In other words, an even function has symmetry with respect to the y-axis.

Use the property of the local minimum

In the given problem, \(f(c)\) is a local minimum. That means that \(f(c) \leq f(x)\) for all x in the neighborhood of \(c\).

Determine the value of the function at \(-c\)

Using the even function property, we have \(f(-c) = f(c)\). Since \(f(c)\) is a local minimum and \(f(-c)\) has the same value, \(f(-c)\) must be a local minimum as well.

Conclusion for even functions

For an even function \(f\), if there is a local minimum at \(c\), then there will also be a local minimum, not a local maximum, at \(-c\). #b. Analyzing odd functions#

Remember the definition of odd functions

An odd function is a function that satisfies the equation \(f(-x) = -f(x)\). In other words, an odd function has symmetry with respect to the origin.

Use the property of the local minimum

In the given problem, \(f(c)\) is a local minimum. That means that \(f(c) \leq f(x)\) for all x in the neighborhood of \(c\).

Determine the value of the function at \(-c\)

Using the odd function property, we have \(f(-c) = -f(c)\). Since \(f(c)\) is a local minimum, \(f(-c)\) must have the opposite value and be a local maximum.

Conclusion for odd functions

For an odd function \(f\), if there is a local minimum at \(c\), then there will be a local maximum, not a local minimum, at \(-c\).

Key concepts

Even Function

An even function is characterized by its symmetry around the y-axis. This means that for any value of -x, the function produces the same output as it does for x. In mathematical terms, this can be written as \[ f(-x) = f(x) \].
Imagine folding the graph of the function along the y-axis; for an even function, every point on the graph on one side will match perfectly with a point on the other side.
Some key points about even functions include:

• They are symmetrical with the y-axis.
• Examples include quadratic functions like \( f(x) = x^2 \) or cosine waves like \( f(x) = \cos(x) \).
• For any local minimum at a point \( c \), the point \( -c \) will also be a local minimum.

Understanding the concept of even functions helps you predict behavior and symmetries in various mathematical contexts.

Odd Function

Odd functions are defined by their symmetry with respect to the origin. This unique characteristic of odd functions means that if you take a point (x, f(x)) on the graph of the function, turning it 180 degrees about the origin will line up perfectly with the point (-x, -f(x)). Mathematically, this property is shown as \[ f(-x) = -f(x) \].
Let's explore some important aspects of odd functions:

• They demonstrate rotational symmetry around the origin.
• Visual examples include functions like \( f(x) = x^3 \) or the sine function \( f(x) = \sin(x) \).
• If a local minimum occurs at a point \( c \), then a local maximum will appear at \( -c \), due to the nature of the directional change at these points.

Recognizing odd functions can significantly help in graph sketching and solving equations that involve symmetry around the origin.

Local Minimum

A local minimum is a point where a function's value is lower than all others in its immediate vicinity. This means that if you were walking on the graph of the function, a local minimum would be like a small valley or dip.
Key aspects of a local minimum include:

• A local minimum occurs at a point \( c \) if \( f(c) \leq f(x) \) for all x near \( c \).
• It's not necessarily the lowest point in the entire graph (a global minimum).
• For even functions, if there's a local minimum at \( c \), the point \( -c \) is also a local minimum due to symmetry.
• For odd functions, a local minimum at \( c \) results in a local maximum at \( -c \).

Understanding local minimum points can significantly aid in graph analysis, optimizations, and understanding the behavior of a function in its specific neighborhood.

Local Maximum

A local maximum is found where a function reaches a peak value compared to nearby points, similar to the top of a hill in a landscape. This concept is crucial because it helps identify where the function temporarily stops increasing before it starts to decrease.
Some important characteristics of a local maximum include:

• A local maximum at a point \( c \) means that \( f(c) \geq f(x) \) for values of x close to \( c \).
• This point might not be the absolute highest point on the graph (a global maximum).
• For odd functions, if a local minimum occurs at \( c \), a local maximum will appear at \( -c \) due to the nature of reflection.

Understanding local maxima is valuable for tasks involving function optimization, determining turning points, and examining the function's behavior in certain regions.