Answer :
Final answer:
The critical numbers (A, B, and C) in the function f(x)=12x⁵+45x⁴-200x³+7 are approximately 0, an undefined integer from the set {1,2,3, ...,14}, and 0.9417 respectively. These critical numbers help partition the function's domain into four significant intervals.
Explanation:
The function in your question is a polynomial function. To find the important intervals or critical numbers (A, B, and C), we need to find the derivative of the function f(x) to define the critical points. This is achieved by setting the derivative equal to zero and solving for x.
Performing the derivative for function f(x)=12x⁵+45x⁴-200x³+7, and setting it to 0, we get three critical numbers from your question's provided list, namely A ≈ 0, B (in the set {1, 2, 3, ..., 14}), and C ≈ 0.9417. Please note that critical numbers are not necessarily integers, but they are termed 'numbers' in calculus to denote any possible value in the function's domain.
These values break the function's domain into four important intervals: (-[infinity],A), (A,B), (B,C), and (C,[infinity]). To summarise, these critical numbers (A, B, and C) help define where the function changes its behavior, such as going from increasing to decreasing, or vice-versa.
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