For the expression √(40-|x-4|³)-37, the maximum value occurs at x=4, and the minimum value occurs at the endpoints of the given domain, x=0 or x=6, due to properties of the absolute, cubic, and square root functions within the expression.
We're asked to find the maximum and minimum values of the expression √(40-|x-4|³)-37 within the domain 0≤x≤6. This task requires an understanding of absolute values, powers, and the square root function, as well as familiarity with finding extrema of functions.
To solve this, examine the expression inside the square root since the outer structure is monotonically increasing. The absolute value function |x-4|³ has a critical point at x = 4. For values of x < 4, the expression inside the absolute value is negative, and for x ≥ 4, it is positive.
Analyze the expression within the given domain. When x is between 0 and 4, |x-4| will become (4-x), and when x is between 4 and 6, |x-4| will be (x-4). Simplifying both scenarios will help identify the maximum and minimum values of the function.
After plotting or calculating, it becomes evident that the maximum value is attained at the boundary x = 4, while the minimum happens at the endpoints of the domain x = 0 or x = 6.
The maximum value is achieved at x=4, and the minimum value is at the endpoints of the domain (x=0 or x=6), based on how the absolute function behaves in the given range in combination with the cubic and square root functions.