Question

Find the critical numbers of the function f(x) = x2/3(6 - x) and then use a Maple graph of the function to determine if each critical number defines a local maximum or local minimum of the function or neither. (If an answer does not exist, enter DNE.)

Answer 1

The critical numbers of the function
`\( f(x) = x^{(2)/(3)}(6 - x) \)` are x = 0 and x = 4. Using a Maple graph, it can be determined that
`\( x = 0 \)` defines a local maximum, and x = 4 defines a local minimum of the function.

Step-by-step explanation:

To find the critical numbers of the function, we need to find the values of x where the derivative f'(x) is equal to zero or undefined. The function is
`\( f(x) = x^{(2)/(3)}(6 - x) \)`, and its derivative is found using the product rule and the chain rule:

`\[ f'(x) = (2)/(3)x^{-(1)/(3)}(6 - x) - x^{(2)/(3)} \]`

Setting f'(x) equal to zero and solving for x gives the critical numbers. In this case, x = 0 and x = 4 are the critical numbers.

To determine if each critical number defines a local maximum or local minimum, we can use the first derivative test. By analyzing the sign of f'(x) around each critical point, we can infer whether the function is increasing or decreasing in those intervals. For x = 0, f'(x) changes from positive to negative, indicating a local maximum. For x = 4, f'(x) changes from negative to positive, indicating a local minimum.

Understanding the behavior of the derivative around critical points provides valuable information about the local extrema of the original function. In this case, x = 0 represents a local maximum, and x = 4 represents a local minimum based on the sign changes in the derivative.