Problem 73 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=x e^{-x} $$

Step-by-Step Solution

Answer

The function \(f(x) = xe^{-x}\) has a relative maximum at \(x=1\) and no relative minima.

Step 1: Find the first derivative of the function

To find the first derivative of the function \(f(x) = xe^{-x}\), we need to use the product rule. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function: \((uv)' = u'v + uv'\) Let \(u = x\) and \(v = e^{-x}\). Then the first derivative of \(f(x)\) is: \(f'(x) = (x)'\cdot e^{-x} + x \cdot (e^{-x})'\)

Step 2: Calculate the derivatives

To complete step 1, we need to find the derivatives of both \(u\) and \(v\): \((x)' = 1\) \((e^{-x})' = -e^{-x}\) Now, substitute these derivatives back into the expression for the first derivative: \(f'(x) = (1)\cdot e^{-x} + x \cdot (-e^{-x})\) Simplify the expression: \(f'(x) = e^{-x} - xe^{-x}\)

Step 3: Find the critical points

To find the critical points, we need to set the first derivative to zero and solve for \(x\): \(e^{-x} - xe^{-x} = 0\) Factor out \(e^{-x}\): \(e^{-x}(1 - x) = 0\) Since \(e^{-x}\) is never equal to zero, we only need to set the other factor to zero: \(1 - x = 0\) Solve for \(x\): \(x = 1\) So, the critical point is at \(x=1\). At this point, we may have a relative maximum or minimum.

Step 4: Find the second derivative of the function

To find the second derivative of the function, we differentiate the first derivative with respect to \(x\): \(f''(x) = (e^{-x} - xe^{-x})'\) Using the product rule and previously found derivatives: \(f''(x) = -e^{-x} - e^{-x} + xe^{-x}\) Simplify the expression: \(f''(x) = -2e^{-x} + xe^{-x}\)

Step 5: Check concavity at the critical point

Evaluate the second derivative at the critical point \(x = 1\): \(f''(1) = -2e^{-1} + 1 \cdot e^{-1}\) Which simplifies to: \(f''(1) = -e^{-1}\) Since \(f''(1) < 0\), the function is concave down at the critical point, meaning it is a relative maximum.

Step 6: State the result

The function \(f(x) = xe^{-x}\) has a relative maximum at \(x=1\). There are no relative minima for this function.

Key Concepts

Relative Maxima

In calculus, to determine if a function has a relative maximum, we look for points where the function reaches a peak compared to its immediate surroundings. These peaks are known as relative maxima. At these points, the function changes from increasing to decreasing. To find them, we usually use the first and second derivatives of the function.

Here's the step-by-step process:

• First, find the first derivative of the function.
• Identify critical points by setting the first derivative to zero.
• Use the second derivative to determine if these critical points are maxima.

In our exercise, the second derivative at the critical point was negative, indicating the function was concave down, confirming it as a relative maximum at that point.

Relative Minima

Relative minima are points on a function where there is a dip—where the function value is lower compared to nearby points. To identify these points, you go through a similar process as finding maxima but look for where the function shifts from decreasing to increasing.

Here's the approach:

• Calculate the first derivative and find critical points by solving for where the derivative is zero.
• Use the second derivative to confirm concavity. If the second derivative is positive, the critical point is a minimum.

In the provided exercise, there were no intervals where the function turned from decreasing to increasing, meaning there were no relative minima present.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It is often represented as \( f'(x) \) and provides us with the slope of the tangent to the curve at any given point.

To find the derivative of a function:

• Use different rules based on the function form, such as the product rule for products of functions.
• Differentiate each part of the function accordingly.

In our example, we used the product rule to differentiate \( xe^{-x} \), yielding \( f'(x) = e^{-x} - xe^{-x} \). This expression then helps in determining where the function reaches its maxima or minima.

Critical Points

Critical points of a function are found where the first derivative is zero or undefined. These points are crucial because they can indicate where the function has relative maxima, relative minima, or saddle points (points that are neither).

Here's how to find and analyze them:

• Set the first derivative equal to zero and solve for \( x \).
• Each solution is a potential critical point.
• Carefully analyze each point using the second derivative to distinguish maxima from minima.

In the exercise, we found the critical point at \( x = 1 \) and used the second derivative to determine that it was a maximum, as the concavity was downward at that point with \( f''(1) = -e^{-1} \).