Final answer:
The critical numbers of the function f(x) = x⁶(x - 3)⁵ are 0 and 3. Both are points where the function reaches a local minimum due to the even powers of x in the factors. b) Critical numbers: 0, 3; Behavior: Local maximum at 0, local minimum at 3 is correct answer.
Step-by-step explanation:
To find the critical numbers of the function f(x) = x⁶(x - 3)⁵, we need to find the values of x where the derivative f'(x) is zero or undefined. The derivative will be zero when the product of the derivatives of x⁶ and (x - 3)⁵ is zero. In this case, both factors are always positive or zero for all real numbers so the original factors must be considered.
The function x⁶ equals zero when x = 0, and the function (x - 3)⁵ equals zero when x = 3. Therefore, the critical numbers are 0 and 3. To determine the behavior at these numbers, we look at the powers of x.
Both powers are even, which means the function will be at a local minimum at both critical numbers. The function doesn't change signs around these numbers due to the even exponents.
The correct answer is therefore: Critical numbers: 0, 3; Behavior: Local minimum at both 0 and 3.