Local Maxima Of F(x) = -e^x(x+4) A Step-by-Step Solution

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Introduction: Delving into the Realm of Local Maxima

In the vast landscape of calculus, finding local maxima is a fundamental pursuit. Local maxima, often referred to as relative maxima, represent the highest points of a function within a specific neighborhood. They are the peaks of the function's graph, offering valuable insights into its behavior and characteristics. In this comprehensive guide, we embark on a journey to uncover the local maximum/maxima of the function f(x) = -e^x(x+4). Our exploration will involve a combination of analytical techniques and insightful discussions, providing a thorough understanding of the process involved.

To effectively navigate this mathematical terrain, we'll first lay the groundwork by defining what constitutes a local maximum. A point c is considered a local maximum of a function f(x) if there exists an open interval around c such that f(c) is greater than or equal to f(x) for all x within that interval. In simpler terms, it's a point that stands taller than its immediate surroundings. The quest for local maxima often involves identifying critical points, which are points where the function's derivative is either zero or undefined. These critical points serve as potential candidates for local maxima, minima, or saddle points. Understanding the relationship between critical points and local extrema is paramount in this endeavor.

The function we'll be dissecting, f(x) = -e^x(x+4), presents an intriguing blend of exponential and linear components. The exponential term, e^x, exhibits continuous growth, while the linear term, (x+4), introduces a linear variation. The interplay between these components shapes the function's behavior, giving rise to the potential for local maxima and minima. Our mission is to pinpoint these critical points and discern whether they correspond to local maxima. We'll employ the power of differential calculus, specifically the first and second derivative tests, to guide our exploration. By analyzing the sign changes of the first derivative and the concavity indicated by the second derivative, we'll be able to confidently identify the local maximum/maxima of the function. So, let's embark on this mathematical adventure and unveil the hidden peaks of f(x) = -e^x(x+4).

Unveiling the Methodology: A Step-by-Step Approach

Embarking on the quest to find the local maximum/maxima of the function f(x) = -e^x(x+4) requires a structured methodology. We will employ a step-by-step approach, leveraging the principles of differential calculus to guide our exploration. This methodology encompasses the following key steps:

  1. Determine the First Derivative: The cornerstone of our approach lies in finding the first derivative of the function, denoted as f'(x). The first derivative provides crucial information about the function's slope at any given point. It tells us whether the function is increasing, decreasing, or stationary. To find the first derivative, we'll employ the product rule, a fundamental differentiation technique. The product rule states that the derivative of the product of two functions is equal to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function. Applying this rule to f(x) = -e^x(x+4) will yield the expression for f'(x).

  2. Identify Critical Points: Critical points are the linchpins in the search for local extrema. These are the points where the function's derivative is either zero or undefined. Critical points represent potential locations where the function changes its direction, transitioning from increasing to decreasing or vice versa. To find the critical points, we'll set the first derivative, f'(x), equal to zero and solve for x. The solutions obtained will be the x-coordinates of the critical points. Additionally, we need to consider points where the derivative is undefined, although this is less common for functions like f(x) = -e^x(x+4).

  3. Apply the First Derivative Test: The first derivative test is a powerful tool for classifying critical points. It involves analyzing the sign of the first derivative in the intervals surrounding the critical points. If the first derivative changes its sign from positive to negative at a critical point, it indicates a local maximum. Conversely, if the first derivative changes its sign from negative to positive, it suggests a local minimum. If the first derivative does not change its sign, the critical point may be a saddle point or neither a maximum nor a minimum. By carefully examining the sign changes of f'(x) around each critical point, we can determine whether it corresponds to a local maximum.

  4. Employ the Second Derivative Test (Optional): The second derivative test provides an alternative method for classifying critical points. It involves finding the second derivative of the function, denoted as f''(x), and evaluating it at the critical points. If the second derivative is negative at a critical point, it indicates a local maximum. If the second derivative is positive, it suggests a local minimum. If the second derivative is zero, the test is inconclusive, and we may need to revert to the first derivative test or other methods. While the second derivative test can be efficient in some cases, it's essential to note that it's not always applicable. For instance, if the second derivative is zero at a critical point, it doesn't provide definitive information.

By meticulously following these steps, we can systematically identify the local maximum/maxima of the function f(x) = -e^x(x+4). Each step builds upon the previous one, providing a clear and logical path towards the solution. Let's now embark on the execution of this methodology, unraveling the hidden peaks of our function.

Step-by-Step Execution: Unraveling the Solution

Having established the methodology, we now embark on the execution phase, meticulously applying each step to the function f(x) = -e^x(x+4). This hands-on process will lead us to the identification of the local maximum/maxima of the function.

  1. Determining the First Derivative:

To find the first derivative, f'(x), we employ the product rule. Recall that the product rule states that if we have two functions, u(x) and v(x), then the derivative of their product is given by:

(u(x)v(x))' = u'(x)v(x) + u(x)v'(x)

In our case, we can identify u(x) = -e^x and v(x) = (x+4). Let's find their respective derivatives:

u'(x) = -e^x

v'(x) = 1

Now, applying the product rule, we get:

f'(x) = u'(x)v(x) + u(x)v'(x)

f'(x) = (-e^x)(x+4) + (-e^x)(1)

f'(x) = -e^x(x+4) - e^x

We can simplify this expression by factoring out -e^x:

f'(x) = -e^x(x+4+1)

f'(x) = -e^x(x+5)

Thus, the first derivative of f(x) is f'(x) = -e^x(x+5).

  1. Identifying Critical Points:

Critical points occur where the first derivative is either zero or undefined. Let's set f'(x) = 0 and solve for x:

-e^x(x+5) = 0

Since e^x is never zero for any real value of x, we can focus on the factor (x+5):

x+5 = 0

x = -5

Therefore, we have one critical point at x = -5. Additionally, we need to consider points where the derivative is undefined. However, in this case, f'(x) = -e^x(x+5) is defined for all real values of x. So, we have only one critical point to consider.

  1. Applying the First Derivative Test:

To apply the first derivative test, we need to analyze the sign of f'(x) in the intervals surrounding the critical point x = -5. This involves choosing test values in the intervals (-∞, -5) and (-5, ∞) and evaluating f'(x) at these test values.

Let's choose x = -6 as a test value in the interval (-∞, -5):

f'(-6) = -e^(-6)(-6+5) = -e^(-6)(-1) = e^(-6)

Since e^(-6) is positive, f'(x) is positive in the interval (-∞, -5). This indicates that the function is increasing in this interval.

Now, let's choose x = -4 as a test value in the interval (-5, ∞):

f'(-4) = -e^(-4)(-4+5) = -e^(-4)(1) = -e^(-4)

Since -e^(-4) is negative, f'(x) is negative in the interval (-5, ∞). This indicates that the function is decreasing in this interval.

The first derivative test reveals that f'(x) changes its sign from positive to negative at x = -5. This signifies that the function has a local maximum at x = -5.

  1. Employing the Second Derivative Test (Optional):

To apply the second derivative test, we first need to find the second derivative, f''(x). We differentiate f'(x) = -e^x(x+5) using the product rule again:

f'(x) = -e^x(x+5)

Let u(x) = -e^x and v(x) = (x+5). Then,

u'(x) = -e^x

v'(x) = 1

Applying the product rule:

f''(x) = u'(x)v(x) + u(x)v'(x)

f''(x) = (-e^x)(x+5) + (-e^x)(1)

f''(x) = -e^x(x+5) - e^x

f''(x) = -e^x(x+5+1)

f''(x) = -e^x(x+6)

Now, we evaluate f''(-5):

f''(-5) = -e^(-5)(-5+6) = -e^(-5)(1) = -e^(-5)

Since f''(-5) = -e^(-5) is negative, the second derivative test confirms that there is a local maximum at x = -5.

  1. Determining the Local Maximum Value:

To find the local maximum value, we substitute x = -5 into the original function f(x) = -e^x(x+4):

f(-5) = -e^(-5)(-5+4) = -e^(-5)(-1) = e^(-5)

Therefore, the local maximum value of the function is e^(-5).

Conclusion: The Pinnacle Unveiled

Through the application of differential calculus, we have successfully identified the local maximum of the function f(x) = -e^x(x+4). Our journey began with defining the concept of local maxima and establishing a structured methodology. We then meticulously executed each step, from determining the first derivative to employing the first and second derivative tests. The culmination of our efforts revealed that the function attains a local maximum at x = -5, with a corresponding value of e^(-5). This point represents the peak of the function within its immediate vicinity, providing valuable insight into its behavior.

The process of finding local maxima is not merely a mathematical exercise; it has profound implications in various fields. In optimization problems, local maxima (and minima) represent the optimal solutions within specific constraints. In economics, they can signify points of maximum profit or minimum cost. In physics, they can correspond to stable equilibrium states. The ability to identify and analyze local extrema is therefore a crucial skill in a wide range of disciplines. As we conclude this exploration, we hope that this comprehensive guide has equipped you with the knowledge and tools necessary to confidently navigate the world of local maxima and unlock their potential in your own endeavors.

  • The function f(x) = -e^x(x+4) has a local maximum at x = -5.
  • The local maximum value is f(-5) = e^(-5).