Let $f(x)=x E^ -2x}$.What Is The Absolute Maximum Value Of $f$?Choose 1 Answer A. $\frac{1 {e}$ B. $ 1 2 E \frac{1}{2e} 2 E 1 [/tex] C. $\frac{1}{e^2}$ D. $f$ Has No Maximum Value

Mar 9, 2025 by ADMIN 195 views

Finding the Absolute Maximum Value of a Function

In calculus, finding the absolute maximum value of a function is a crucial problem that has numerous applications in various fields, including physics, engineering, and economics. The absolute maximum value of a function is the largest value that the function attains over its entire domain. In this article, we will explore how to find the absolute maximum value of the function $f(x) = x e^{-2x}$ .

The given function is $f(x) = x e^{-2x}$ . This function is a product of two functions: $x$ and $e^{-2x}$ . The function $e^{-2x}$ is an exponential function that decreases as $x$ increases, while the function $x$ increases as $x$ increases. To find the absolute maximum value of $f(x)$ , we need to analyze the behavior of the function as $x$ varies.

To find the absolute maximum value of $f(x)$ , we need to find the critical points of the function. Critical points are the points where the function changes from increasing to decreasing or from decreasing to increasing. To find the critical points, we need to find the derivative of the function and set it equal to zero.

Let's find the derivative of $f(x)$ using the product rule:

f'(x) = \frac{d}{dx} (x e^{-2x})

Using the product rule, we get:

f'(x) = e^{-2x} + x (-2) e^{-2x}

Simplifying the expression, we get:

f'(x) = e^{-2x} - 2x e^{-2x}

Now, we need to set the derivative equal to zero and solve for $x$ :

e^{-2x} - 2x e^{-2x} = 0

Factoring out $e^{-2x}$ , we get:

e^{-2x} (1 - 2x) = 0

Since $e^{-2x}$ is never equal to zero, we can set $1 - 2x = 0$ and solve for $x$ :

1 - 2x = 0

Solving for $x$ , we get:

x = \frac{1}{2}

Therefore, the critical point of the function is $x = \frac{1}{2}$ .

Now that we have found the critical point, we need to analyze its behavior. To do this, we need to find the second derivative of the function and evaluate it at the critical point.

Let's find the second derivative of $f(x)$ :

f''(x) = \frac{d}{dx} (e^{-2x} - 2x e^{-2x})

Using the product rule and the chain rule, we get:

f''(x) = -2 e^{-2x} + e^{-2x} - 2x (-2) e^{-2x}

Simplifying the expression, we get:

f''(x) = -2 e^{-2x} + e^{-2x} + 4x e^{-2x}

Now, we need to evaluate the second derivative at the critical point $x = \frac{1}{2}$ :

f''(\frac{1}{2}) = -2 e^{-1} + e^{-1} + 2 e^{-1}

Simplifying the expression, we get:

f''(\frac{1}{2}) = e^{-1}

Since the second derivative is positive at the critical point, we can conclude that the function is concave up at the critical point.

In conclusion, we have found the absolute maximum value of the function $f(x) = x e^{-2x}$ . The critical point of the function is $x = \frac{1}{2}$ , and the second derivative is positive at this point, indicating that the function is concave up. Therefore, the absolute maximum value of the function is attained at the critical point $x = \frac{1}{2}$ .

To find the absolute maximum value of the function, we need to evaluate the function at the critical point:

f(\frac{1}{2}) = \frac{1}{2} e^{-1}

Simplifying the expression, we get:

f(\frac{1}{2}) = \frac{1}{2e}

Therefore, the absolute maximum value of the function is $\frac{1}{2e}$ .

The absolute maximum value of the function $f(x) = x e^{-2x}$ is $\boxed{\frac{1}{2e}}$ .

The problem of finding the absolute maximum value of a function is a fundamental problem in calculus. In this article, we have explored how to find the absolute maximum value of the function $f(x) = x e^{-2x}$ . We have found the critical point of the function, analyzed its behavior, and concluded that the absolute maximum value of the function is attained at the critical point. The absolute maximum value of the function is $\frac{1}{2e}$ .

[1] Calculus, 3rd edition, Michael Spivak
[2] Calculus, 2nd edition, James Stewart
[3] Calculus, 1st edition, Michael Spivak
Q&A: Finding the Absolute Maximum Value of a Function

In our previous article, we explored how to find the absolute maximum value of the function $f(x) = x e^{-2x}$ . We found the critical point of the function, analyzed its behavior, and concluded that the absolute maximum value of the function is attained at the critical point. In this article, we will answer some frequently asked questions related to finding the absolute maximum value of a function.

A: The absolute maximum value of a function is the largest value that the function attains over its entire domain.

A: To find the absolute maximum value of a function, you need to find the critical points of the function, analyze their behavior, and evaluate the function at these points.

A: A critical point is a point where the function changes from increasing to decreasing or from decreasing to increasing.

A: To find the critical points of a function, you need to find the derivative of the function and set it equal to zero.

A: The second derivative is the derivative of the first derivative of a function.

A: You need to find the second derivative to analyze the behavior of the function at the critical points.

A: The concavity of a function is the shape of the function's graph. If the function is concave up, it means that the function is increasing at an increasing rate.

A: You can determine the concavity of a function by finding the second derivative and evaluating it at the critical points.

A: The absolute maximum value of the function $f(x) = x e^{-2x}$ is $\frac{1}{2e}$ .

A: To find the absolute maximum value of a function using calculus, you need to follow these steps:

Find the derivative of the function.
Set the derivative equal to zero and solve for the critical points.
Find the second derivative and evaluate it at the critical points.
Analyze the behavior of the function at the critical points.
Evaluate the function at the critical points to find the absolute maximum value.

In conclusion, finding the absolute maximum value of a function is a crucial problem in calculus. In this article, we have answered some frequently asked questions related to finding the absolute maximum value of a function. We have also provided a step-by-step guide on how to find the absolute maximum value of a function using calculus.

[1] Calculus, 3rd edition, Michael Spivak
[2] Calculus, 2nd edition, James Stewart
[3] Calculus, 1st edition, Michael Spivak