Final answer:
The critical number A for the function f(x) = 7x + 2x⁻¹ is A = √(2/7). The function is not defined at B, where B = 0, and there is no additional critical number C.
Step-by-step explanation:
To determine the critical numbers A and C of the function f(x) = 7x + 2x⁻¹, we first need to find where the derivative of the function is equal to zero or undefined, as these points can indicate where the function changes its behavior. The derivative of f(x) is f'(x) = 7 - 2x⁻². Setting the derivative to zero gives us f'(x) = 0 → 7 - 2x⁻² = 0 → x = ± √(2/7). Since x cannot be negative (as indicated by the original function f(x) which has a term with x⁻¹), we only consider the positive root. Therefore, A = √(2/7).
For B, we find where the function is undefined. Given that the term 2x⁻¹ is in the function, f(x) will be undefined when x = 0. So, B = 0. Next, we look for another point where the derivative equals zero or the function is undefined, but since the root we found is the only critical point from the derivative and the function only has one point of undefined value at x = 0, there are no further critical values to find, thus no additional value for C.
The intervals of importance for this function, based on the critical number found and the point of undefined value of the function, are (-∞, √(2/7)), [√(2/7), 0), (0, ∞).