Question

Consider the function f(x)=12x3+15x4−240x3+7. For this function, there are four important intervals: (−[infinity],A],[A,B],[B,C], and(C,[infinity]) where A, B, and C are certain values.

Please provide the specific values of A, B, and C that define these intervals.

Answer 1

The specific values of A, B, and C that define the intervals (-∞, A], [A, B], [B, C], and (C, ∞) for the function f(x)=12x^3+15x^4-240x^3+7 are 0, 4/5, and ∞, respectively.

Step-by-step explanation:

The function f(x)=12x^3+15x^4-240x^3+7 can be divided into four important intervals: (-∞, A], [A, B], [B, C], and (C, ∞). To find the values of A, B, and C, we need to determine the critical points of the function by taking its derivative and setting it equal to zero. By differentiating f(x) and setting it equal to zero, we get 36x^2 + 60x^3 - 240x^2 = 0. Factoring out common terms, we get 12x^2(3 + 5x - 20) = 0. Setting each factor equal to zero, we find the critical points: x = 0 and x = 4/5. Therefore, the specific values of A, B, and C are 0, 4/5, and ∞, respectively.