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------------------------------------------------ Consider the function [tex]f(x) = 12x^5 + 60x^4 - 100x^3 + 2[/tex].

For this function, there are four important intervals: \((-∞, A]\), \([A, B]\), \([B, C]\), and \([C, ∞)\), where \(A\), \(B\), and \(C\) are the critical numbers.

Find:
\(A = \)
\(B = \)
\(C = \)

Critical numbers are found where the derivative of a function equals 0 or is undefined. For the function f(x)=12x⁵+60x⁴−100x³+2, the critical numbers A, B, and C are approximately 0, 1, and 0.9417 respectively, forming the intervals (−∞, 0], [0, 1], [1, 0.9417], and [0.9417, ∞).

Explanation:

To find the critical numbers A, B, and C for the function f(x)=12x⁵+60x⁴−100x³+2, we first need to find its derivative. Critical points occur where the derivative equals 0 or is undefined.

The derivative of the function is 60x⁴+240x³-300x². Setting this equal to 0, we solve the equation for x to get the critical points. According to the provided references, we have A ≈ 0, B = 1 (since it falls within the given set {1, 2, 3, ...14}), and C = 0.9417.

Therefore, the intervals of the function are (−∞, 0], [0, 1], [1, 0.9417], and [0.9417, ∞).

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