Question

(a) use a computer algebra system to graph the function and approximate any absolute extrema on the indicated 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). $$ f(x)=(x-4) \arcsin \frac{x}{4} $$

Short answer

Compute the function's graph and establish its absolute extremes. Then derive the critical numbers using calculus. Evaluate the original function at each critical number and the endpoints to find any absolute extremal points. Then compare the results from part (a) graphing and part (b) calculus.

Step-by-step solution

Graph the function

Using graphing software, plot the given function \(f(x) = (x-4) \arcsin (\frac{x}{4})\) on the real number line.

Approximate absolute extrema

Analyze the plotted graph to approximately identify any absolute maximum and minimum points. These are the highest and lowest points on the graph over the entire domain.

Find the function's derivative

Differentiate the function to find its derivative. The derivative of a function can help identify critical points on the graph. The derivative of the given function could be found using the product rule and the chain rule.

Find the critical numbers

The critical numbers of a function are where its derivative equals zero or is undefined. After finding the derivative in the previous step, set the derivative equal to zero and solve for \(x\) to get the critical numbers.

Find the absolute extrema using critical numbers

Evaluate the original function at the critical numbers and endpoints. The greatest of these values is the absolute maximum, and the least of these values is the absolute minimum. Identify any absolute extrema not located at the endpoints.

Compare the results

Compare the results from part (a) and part (b) to verify that they match. They should fall in the same locations on the graph.