High School

Consider the function [tex]f(x) = 12x^5 + 45x^4 - 360x^3 + 6[/tex]. For this function, there are four important intervals: [tex](-\infty, A][/tex], [tex][A, B][/tex], [tex][B, C][/tex], and [tex][C, \infty)[/tex], where A, B, and C are the critical numbers.

Find A: _________

Answer :

A = -6 as the point of local minimum.

To determine the critical number A for the function f(x)=12x⁵+45x⁴-360x³+6, we need to find the derivative of f(x) and solve for the points where the derivative equals zero.

  1. The derivative of f(x) is f'(x) = 60x⁴ + 180x³ - 1080x².
  2. Setting f'(x) to zero: 60x⁴ + 180x³ - 1080x² = 0.
  3. Factor the equation: 60x²(x² + 3x - 18) = 0.
  4. Solve for x: x = 0 or x² + 3x - 18 = 0.
  5. Solving the quadratic equation: (x + 6)(x - 3) = 0, so x = -6, 3.
  6. The critical numbers are x = 0, -6, and 3. From the given reference, we know that A is the critical number associated with a local minimum, which is x = -6.

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