Problem 65
Question
Odd functions If an odd function \(g(x)\) has a local minimum value at \(x=c\) , can anything be said about the value of \(g\) at \(x=-c\) ? Give reasons for your answer.
Step-by-Step Solution
Verified Answer
At \(x = -c\), \(g\) has a local maximum and \(g(-c) = -g(c)\).
1Step 1: Understanding Odd Functions
An odd function is defined by the property that for all values of \(x\), \(g(-x) = -g(x)\). This means that the graph of \(g(x)\) is symmetric with respect to the origin. We need to see what this symmetry implies if the function has a local minimum at a point \(x = c\).
2Step 2: Local Minimum Definition
A local minimum at \(x = c\) means that \(g(c)\) is less than or equal to \(g(x)\) for points \(x\) near \(c\). Thus, \(g(c)\) represents the smallest value of the function in a small neighborhood around \(c\).
3Step 3: Analyzing Symmetry at \(x = -c\)
Since \(g(x)\) is odd, and \(g(-c) = -g(c)\), the value at \(x = -c\) will be the negative of the value at \(x = c\). This symmetry implies that if \(g(c)\) is a local minimum, then \(g(-c)\) must be a local maximum because the negative of a local minimum is a local maximum.
Key Concepts
Function SymmetryLocal MinimumLocal MaximumFunction Properties
Function Symmetry
Functions can have interesting symmetrical characteristics that shape their graphs and properties. One common type is **odd functions**. An odd function satisfies the condition \( g(-x) = -g(x) \) for all \( x \). This means that its graph is symmetric about the origin. In simpler terms, if you rotate the graph 180 degrees around the origin, it will look the same. This property directly affects how we perceive and calculate other features of the function, like local minima and maxima.
Local Minimum
A local minimum of a function occurs when a particular value of the function is lower than all other values in the immediate vicinity. Specifically, at a point \( x = c \), if \( g(c) \leq g(x) \) for nearby points \( x \), we have a local minimum. Local minima can be thought of as valleys in the graph of the function. However, in odd functions, calculating minima can reveal more insight due to their symmetrical nature, impacting corresponding points in different quadrants of the graph.
Local Maximum
Interestingly, when discussing odd functions and their local minima, a phenomenon occurs due to symmetry: the existence of a local minimum at \( x = c \) implies a local maximum at \( x = -c \). This arises from the rule \( g(-c) = -g(c) \). If \( g(c) \) is the lowest value in its vicinity, \( -g(c) \) will naturally be the highest value in the vicinity of \( x = -c \), termed as a local maximum. This shows how symmetry can dynamically transform the understanding of a function's local behavior.
Function Properties
Odd functions carry unique properties inherently tied to their symmetry. By definition, \( g(-x) = -g(x) \) must hold for every point and this causes the graph to exhibit peculiar behaviors.
- Origin Symmetry: The graph mirrors itself through the origin.
- Local Mirroring: Local minima convert to local maxima at opposite points of symmetry.
- Transformation Consistency: Rotational and reflective symmetry assure predictable transformations.
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