Problem 31 - Calculus Early Transcendentals

Question

Use a graphing utility to estimate the absolute maximum and minimum values of \(f \), if any, on the stated interval, and then use calculus methods to find the exact values. \(f(x)=x^{2 / 3}(20-x) ;[-1,20]\)

Step-by-Step Solution

Answer

The absolute maximum is approximately 21.544 at \(x=10\), and the minimum is 0 at \(x=20\).

Step 1: Graphing the Function

Use a graphing utility, such as Desmos or a graphing calculator, to plot the function \(f(x) = x^{2/3}(20-x)\) over the interval \([-1, 20]\). This visual representation will help us estimate the points where the function reaches its absolute maximum and minimum on the given interval.

Step 2: Identifying Critical Points

To find the critical points, differentiate the function. First, find \(f'(x)\) using the product rule for derivatives: \(f(x) = x^{2/3}(20-x)\) implies \(f'(x) = (2/3)x^{-1/3}(20-x) + x^{2/3}(-1)\). Set \(f'(x) = 0\) and solve for \(x\).

Step 3: Solving for Critical Points

Solve the derivative equation \(\frac{2}{3}x^{-1/3}(20-x) - x^{2/3} = 0\) resulting in the equation \(\frac{2}{3}(20-x) = x\). Simplify to find potential critical points: rearranging gives \(x = 10\).

Step 4: Evaluating Endpoints and Critical Points

Evaluate \(f(x)\) at the critical point \(x = 10\) and at the endpoints of the interval \(x = -1\) and \(x = 20\). Calculate \(f(-1), f(10), \text{ and } f(20)\).

Step 5: Calculating Function Values

Calculate \(f(-1) = (-1)^{2/3}(20 - (-1)) = 21\), \(f(10) = 10^{2/3}(20 - 10) = 10^{2/3} \times 10 = 10^{5/3}\), and \(f(20) = 20^{2/3}(20-20) = 0\).

Step 6: Comparing Values

Compare the values of \(f(-1) = 21\), \(f(10) = 10^{5/3}\), and \(f(20) = 0\). From these, identify the maximum and minimum values of \(f\) on the interval \([-1, 20]\). Given \(10^{5/3} \approx 21.544\), \(f(10)\) is the maximum and \(f(20)\) is the minimum.

Key Concepts

Graphing Utility

A graphing utility is a tool that helps visualize mathematical functions. Such tools include graphing calculators and online platforms like Desmos.
These utilities display functions as graphs, making it easier to estimate important points such as maximums and minimums.

• To use a graphing utility, enter the function into the program. In our exercise, this means inputting the function \(f(x) = x^{2/3}(20-x)\).
• The graph is displayed over a specified interval, in this case, \([-1, 20]\).
• Visually analyzing the graph allows students to see where the function reaches its highest and lowest points.

Using a graphing utility provides a comprehensive overview before diving into more detailed calculus methods.

Critical Points

Critical points are where a function’s derivative is zero or undefined. These points can indicate where a function reaches its local maximum or minimum.
To find these for \(f(x) = x^{2/3}(20-x)\):

• Start by differentiating the function. For this exercise, use the product rule since the function is a product of two expressions.
• Find \(f'(x)\) and solve \(f'(x) = 0\).
• Solving tells us potential points where the graph 'turns' or levels out, such as in this example, where the critical point was found to be \(x = 10\).

Recognizing critical points is crucial for identifying behavior changes in a function.

Product Rule

The product rule is a fundamental technique in calculus used for differentiating functions that are the product of two simpler functions.
The rule is defined as: if \(u(x)\) and \(v(x)\) are functions, then the derivative of their product is \(uv' + vu'\).

• For the function \(f(x) = x^{2/3}(20-x)\), identify the parts: \(u(x) = x^{2/3}\) and \(v(x) = (20-x)\).
• Differentiate these parts separately: \(u'(x) = \frac{2}{3}x^{-1/3}\) and \(v'(x) = -1\).
• Apply the product rule: \(f'(x) = u(x)v'(x) + v(x)u'(x)\).

This method simplifies finding derivatives of more complicated expressions by breaking them down into manageable pieces.

Maximum and Minimum Values

Determining maximum and minimum values involves evaluating the function at critical points and endpoints within a given interval.
For the function \(f(x) = x^{2/3}(20-x)\) in interval \([-1, 20]\):

• Evaluate \(f(x)\) at the critical point found (\(x = 10\)) and endpoints (\(x = -1\) and \(x = 20\)).
• Calculate \(f(-1) = 21\), \(f(10) \approx 21.544\), and \(f(20) = 0\).
• Compare these values: \(f(10)\) is the absolute maximum, and \(f(20)\) is the minimum on the interval.

By estimating graphically and confirming with calculus, we ensure comprehensive understanding of a function's behavior across its domain.