Problem 8 - Calculus Single Variable

Question

In each of Exercises \(7-22,\) use Fermat's Theorem to locate each \(c\) for which \(f(c)\) is a candidate extreme value of the given function \(f\) $$ f(x)=12 x^{2}+48 x $$

Step-by-Step Solution

Answer

The candidate point for an extreme value is \(c = -2\).

Step 1: Understand Fermat's Theorem

Fermat's Theorem states that if a function has a local extremum (maximum or minimum) at a point \(c\), and if the derivative exists at \(c\), then the derivative at that point is zero: \(f'(c) = 0\). The extremum points are potential candidates for maxima or minima.

Step 2: Differentiate the Function

Find the derivative of the given function \(f(x) = 12x^2 + 48x\). Use the power rule for differentiation, which gives \(f'(x) = 2 \cdot 12x + 48 = 24x + 48\).

Step 3: Set the Derivative to Zero

According to Fermat's Theorem, set the derivative \(f'(x)\) equal to zero to find the critical points. Solve the equation: \(24x + 48 = 0\).

Step 4: Solve for x

Solve the equation \(24x + 48 = 0\) to find the value of \(x\). Subtract 48 from both sides to obtain \(24x = -48\). Divide both sides by 24 to isolate \(x\), resulting in \(x = -2\).

Step 5: Interpret the Result

The solution \(x = -2\) indicates the point \(c\) where the derivative equals zero, and hence a candidate for a local extremum. By locating this point, we find where the extreme values could occur.

Key Concepts

Extreme Values

Extreme values in a function are the largest or smallest values that the function can take. These are often referred to as maxima (the highest values) and minima (the lowest values). Understanding where these values occur can help in analyzing the behavior of a function. When a function reaches an extreme value, it occurs at specific points called extrema. These include:

• Global maxima and minima - the overall highest and lowest points of the function across its entire domain.
• Local maxima and minima - the highest and lowest points within a particular neighborhood, but not necessarily across the entire domain.

Fermat's Theorem plays a crucial role in identifying potential points for local extrema, where the derivative of the function equals zero. Finding these points helps narrow down the search for extreme values.

Derivative

The derivative of a function provides significant insights into the function's behavior. It measures how a function's output changes with respect to changes in the input. For the function given, \(f(x) = 12x^2 + 48x\), the derivative calculated using the power rule becomes \(f'(x) = 24x + 48\). Key points to remember about derivatives:

• They tell us the slope, or rate of change, of the function at any given point.
• A positive derivative means the function is increasing, while a negative derivative means it is decreasing.
• A zero derivative indicates no change in the function's slope at that point.

By differentiating a function, critical points where extreme values might occur can be located.

Critical Points

Critical points of a function occur where its derivative is zero or undefined. To find these points, the derivative of the function, \(f'(x)\), is set to zero and solved for \(x\). In the example used (\(24x + 48 = 0\)), the solution \(x = -2\) represents a critical point.Here are some characteristics of critical points:

• They are potential candidates for extremum points (either maximum, minimum, or point of inflection).
• Not every critical point corresponds to a local extremum; further evaluation, such as using the second derivative test, might be needed.

Identifying critical points is the first step in pinpointing where a function could have extreme values.

Local Extrema

Local extrema are the local maximum and minimum points a function can attain within a certain interval. They are not as broad as global extrema but are crucial in understanding the function's behavior over specific sections.To identify local extrema using Fermat's Theorem, these steps are generally followed:

• Find the derivative of the function and solve for when it equals zero to determine critical points.
• Evaluate the behavior around these critical points to see if they represent a local maximum or minimum.

In our example, at \(x = -2\), we would need to examine the function's values before and after this point to decide if \(f(-2)\) is a local maximum or minimum. This exploration provides valuable insight into how a function varies across its domain.