Problem 64 - Essential Calculus Early Transcendentals

Question

If \(f\) has a local minimum value at \(c,\) show that the function \(g(x)=-f(x)\) has a local maximum value at \(c .\)

Step-by-Step Solution

Answer

If \( f \) has a local minimum at \( c \), then \( g(x) = -f(x) \) has a local maximum at \( c \).

Step 1: Understand the Given Information

We are given that the function \( f \) has a local minimum at \( c \). This means that for values of \( x \) close to \( c \), \( f(c) \leq f(x) \).

Step 2: Define the Function \( g(x) \)

We want to analyze the function \( g(x) = -f(x) \) to prove it has a local maximum at \( c \).

Step 3: Analyze the Behavior Before and After \( c \)

Since \( f \) has a local minimum at \( c \), function \( f \) satisfies \( f(c) \leq f(x) \) for \( x \) close to \( c \). This implies that \( -f(c) \geq -f(x) \).

Step 4: Conclude About \( g(x) \)

Since \( -f(c) \geq -f(x) \), the function \( g(x) = -f(x) \) has a local maximum at \( c \) because, by definition, a function reaches a local maximum if it achieves a higher value at \( c \) compared to other values in its vicinity.

Key Concepts

Local Minimum

In calculus, a local minimum of a function occurs at a point where the function takes on a value lower than at any nearby points. When we say a function has a local minimum at \( c \), it means that if you look at the values of the function close to \( c \), the function's value at \( c \) will be less than or equal to the values at nearby points.

For example, imagine you're standing at the lowest point in a small dip in the landscape. This is a local minimum because it's the lowest point in your vicinity, even if there are deeper valleys elsewhere. Mathematically, if \( f \) has a local minimum at \( c \), then for all \( x \) in some small interval around \( c \), \( f(c) \leq f(x) \).

Understanding local minima is crucial when analyzing the behavior of functions, especially when optimizing or finding the points of interest, like turning points in graphs.

Local Maximum

A local maximum occurs when a function takes on a value larger than at any of the nearby points around it. When considering the point \( c \) as a local maximum for some function, \( g(x) \), we mean that \( g(c) \) is larger than or equal to \( g(x) \) for all \( x \) near \( c \).

To visualize, picture yourself on a hilltop. You're at a local maximum because your elevation is higher than everywhere immediately surrounding you, even if there are taller peaks elsewhere around you. In calculus terms, this can be seen by having \( g(c) \geq g(x) \) for \( x \) close to \( c \).

In the exercise, we transform a local minimum of a function \( f \) into a local maximum by considering \( g(x) = -f(x) \). By flipping the function values' sign, a local maximum appears where there was previously a local minimum.

Function Transformation

Function transformation involves modifying a given function to achieve a certain effect, such as shifting its graph or altering its features. One common type of transformation is negating the values of a function, which reflects its graph over the x-axis.

For instance, if you have \( f(x) \) and you modify it to become \( g(x) = -f(x) \), you're essentially flipping the function upside down. This conversion can transform local minima into local maxima, as demonstrated in the exercise.

Here's why it works:

• If \( f(c) \) is a local minimum, it means \( f(c) \leq f(x) \) for \( x \) near \( c \).
• Transforming \( f(x) \) to \( g(x) = -f(x) \) flips the inequality, making \( -f(c) \geq -f(x) \).
• This indicates that \( g(x) \) reaches a local maximum at \( c \), because the negative sign reverses the order, turning the lowest point into a peak.

Function transformations like these are powerful tools in calculus, allowing you to easily manipulate functions' characteristics for various analyses.