Verify that the following functions satisfy the conditions of Theorem 4.9 on their domains. Then find the location and value of the absolute extrema guaranteed by the theorem.


f(x) = x√(3-x)

Verified step by step guidance

1. Identify the domain of the function f(x) = x√(3-x) by determining the values of x for which the expression under the square root is non-negative, i.e., 3 - x ≥ 0.
2. Set the inequality 3 - x ≥ 0 to find the interval for x, which gives x ≤ 3. Therefore, the domain of f(x) is (-∞, 3].
3. Next, find the critical points of the function by taking the derivative f'(x) and setting it equal to zero. Use the product rule for differentiation since f(x) is a product of two functions: x and √(3-x).
4. Solve the equation f'(x) = 0 to find the critical points within the domain. Also, check the endpoints of the domain (x = 3) to evaluate the function.
5. Evaluate the function f(x) at the critical points and the endpoints to determine the absolute maximum and minimum values on the domain.