Problem 57 - Essential Calculus

Question

Let the function \(f\) be differentiable on an interval \(I\) containing c. If \(f\) has a maximum value at \(x=c\), show that \(-f\) has a minimum value at \(x=c\)

Step-by-Step Solution

Answer

The derivative of \(-f(x)\) is \(-f'(x)\). As the maximum of \(f(x)\) at \(x=c\) is a critical point where \(f'(c) = 0\), \(-f(x)\) also has a critical point at \(x=c\) with \(-f'(c) = 0\). Finally, as maximum points under negation become minimum points, we can confirm that \(-f(x)\) has a minimum at \(x=c\).

Step 1: Finding the derivative of \(-f(x)\)

First let's find the derivative of \(-f(x)\). As \(-f(x)\) is simply the negative of \(f(x)\), the derivative \(-f'(x)\) or \(f_1'(x)\) is simply the negative of the derivative of \(f(x)\). This is because the derivative of a constant times a function is the constant times the derivative of the function. So we have \(f_1'(x) = -f'(x)\).

Step 2: Analyze the derivative at \(x=c\)

Let's now analyze the derivative \(f_1'(x)\) at \(x=c\). Given \(f'(c) = 0\), we can substitute in our expression from Step 1 to find \(f_1'(c) = -f'(c) = -0 = 0\). So \(-f(x)\) has a critical point at \(x=c\), just like \(f(x)\).

Step 3: Establish \(-f(c)\) as a minimum

Because \(f(x)\)'s maximum at \(x=c\) became a minimum when we took the negation, \(f(c)\) is a maximum of \(f\) over the interval \(I\). This means that for all \(x\) in \(I\), \(f(c) \geq f(x)\). Therefore, when we take the negation, for all \(x\) in \(I\), \(-f(c) \leq -f(x)\). Thus \(-f(x)\) has its minimum value at \(x=c\), as was to be shown.

Key Concepts

Critical Point

A critical point is where the derivative of a function equals zero or does not exist. It is an important concept in calculus. A function might have a maximum, minimum, or saddle point (which is neither a maximum nor minimum) at a critical point.
To find critical points, you follow these steps:

• Differentiate the function to find its derivative.
• Set the derivative equal to zero to find the potential critical points.
• Analyze whether these points are actually maxima, minima or neither by checking the second derivative or using other methods.

In the context of the given exercise, at the point \(x=c\), the derivative \(f'(c) = 0\), identifying \(x=c\) as a critical point. This occurs because \(f(x)\) has a maximum at this location, leading to a zero derivative, which is a defining characteristic of critical points.

Derivative

The derivative of a function provides the rate at which the function value changes as its input changes. Understanding how to compute and interpret the derivative is crucial in analyzing the behavior of functions.
Here are key points about derivatives:

• The derivative of a function \(f(x)\), denoted as \(f'(x)\) or \(\frac{df}{dx}\), measures the slope of the tangent line to the function's curve at any given point.
• A positive derivative indicates that the function is increasing, and a negative derivative shows it is decreasing at that particular point.
• Zerově derivatives correspond with potential maximum, minimum, or other critical points.Independent from any constants that might multiply the function, the derivative principle holds true as multiplying by any constant simply scales the rate of change.

In this exercise, we see that changing \(f(x)\) to \(-f(x)\) inverts the derivative's result by multiplying it by -1. Therefore, \(-f'(x) = -f_1'(x)\), showing that the critical nature is preserved through negative transformation.

Maximum and Minimum Values

Maximum and minimum values refer to the highest and lowest points on a function within a given interval. These points help analyze the behavior of functions and solve optimization problems.
Recall:

• Local maximum: This point is where the function reaches a peak within a small neighborhood. For example, if \(f(c) \geq f(x)\) for all \(x\) near \(c\), then \(f(c)\) is a local maximum.
• Local minimum: The function hits its lowest point in the immediate area. If \(f(c) \leq f(x)\) for all \(x\) close to \(c\), \(f(c)\) would be a local minimum.

When the exercise describes a function's maximum becoming a minimum through negation, the conversion process involves flipping the inequality signs.
The critical transformation means \(f(c)\) was a local maximum of \(f(x)\), thus \(\-f(c)\) becomes a local minimum of \(-f(x)\), hence directly confirming the function's behavior inversion due to the multiplication by -1.