Problem 46 - Calculus of a Single Variable

Question

Finding Extrema Using Technology In Exercises 45 and \(46,(\text { a) use a computer algebra system to graph the function }\) and approximate any absolute extrema on the given interval. (b) Use the utility to find any critical numbers, and use them to find any absolute extrema not located at the endpoints. Compare the results with those in part (a). $$ f(x)=\frac{4}{3} x \sqrt{3-x}, \quad[0,3] $$

Step-by-Step Solution

Answer

The function \(f(x)=\frac{4}{3} x \sqrt{3-x}\) can have absolute extrema at the endpoints of the interval [0,3] or at the critical numbers in this interval. This can be determined using a combination of technology for graphing and mathematical concepts for evaluating critical numbers.

Step 1: Graph the function

Using a computer algebra system, plot the function \(f(x)=\frac{4}{3} x \sqrt{3-x}\) on the interval [0,3]. Observe the behavior of the function - where it reaches its highest and lowest points.

Step 2: Approximate the extrema

From the graph, identify the points of highest and lowest y-values, which represent the approximate absolute maximum and minimum respectively.

Step 3: Find critical numbers

The critical numbers of a function can be found where the derivative of the function is zero or undefined. Here, compute the derivative of \(f(x)\) and set it equal to zero, solving for \(x\) values. These are the critical numbers.

Step 4: Identify extrema not at endpoints

Evaluate the function at the critical numbers found in the previous step. If the function achieves higher y-values at these points than at the endpoints, then these are absolute maxima not located at the endpoints. Likewise for minima.

Step 5: Compare results

Compare the extrema obtained from the graph and from the critical numbers. They should match, verifying the accuracy of both methods.

Key Concepts

Critical Numbers

Understanding critical numbers is pivotal for finding extrema. Critical numbers of a function occur where its derivative is either zero or undefined. These points are essential because they can be potential locations for local maxima, minima, or points of inflection.
When working through a problem, the process typically involves taking the derivative of the function, setting it equal to zero, and solving for the variable. This step will help you identify the critical numbers.
For instance, if you have a function like \(f(x)=\frac{4}{3} x \sqrt{3-x}\), you would first compute its derivative \(f'(x)\). Then, you set \(f'(x) = 0\) and solve for \(x\) to find the critical numbers.

• These critical numbers might correspond to points where the function has local peak or valley values.
• They could also be places where the function stops increasing or decreasing temporarily.

It's noteworthy that not all critical numbers are points of extrema, as further analysis is needed to confirm their nature.

Absolute Extrema

Finding absolute extrema means determining the highest or lowest points that a function reaches on a specified interval. These are often called the absolute maximum and minimum values. Absolute extrema occur either at critical numbers found within the interval or at the endpoints of the interval.
To identify absolute extrema, evaluate the function at:

• The critical numbers within the interval.
• The endpoints of the interval.

Compare these values to determine the highest and lowest points.
For our function example, \(f(x)=\frac{4}{3} x \sqrt{3-x}\) on \([0,3]\), compute the function's values at each endpoint and at any critical numbers identified.
You may find that the function has its extreme values just at the critical numbers or possibly at the endpoints, depending on how the function behaves across the interval.

Computer Algebra System

Using a Computer Algebra System (CAS) is a modern and efficient way to graph functions and solve mathematical problems involving calculus. These systems can perform symbolic computations like derivatives, integrals, and more.
When tackling a function to find extrema, a CAS can quickly provide a visual graph of the function, making it easier to identify peaks and valleys visually. The graph aids in approximating where the function's absolute maxima and minima might occur.
Using a CAS, you can:

• Automatically compute the derivative of a function to find its critical numbers.
• Graph the function to visually inspect peaks, valleys, and trends over an interval.
• Cross-check analytical findings with graphical observations for added accuracy.

Overall, a Computer Algebra System can greatly enhance your problem-solving process by automating complex calculations and providing visual insights. This tool is especially invaluable when comparing results of manually determined critical points with those observed on a graph.