this table shows some values of the cubic function f(x).\n| x | f(x) |\n| -5 | -2 |\n| -4 | 0 |\n| -2 | -2…

this table shows some values of the cubic function f(x).\n| x | f(x) |\n| -5 | -2 |\n| -4 | 0 |\n| -2 | -2 |\n| -1 | 0 |\n| 0 | 8 |\nwhat is the difference between the relative minimum value of f(x) and the relative minimum value of the function g(x)=(x - 3)^2(x - 6)?\na. 2 units\nb. 3 units\nc. 6 units\nd. 7 units

this table shows some values of the cubic function f(x).\n| x | f(x) |\n| -5 | -2 |\n| -4 | 0 |\n| -2 | -2 |\n| -1 | 0 |\n| 0 | 8 |\nwhat is the difference between the relative minimum value of f(x) and the relative minimum value of the function g(x)=(x - 3)^2(x - 6)?\na. 2 units\nb. 3 units\nc. 6 units\nd. 7 units

Answer

Explanation:

Step1: Encontrar el mínimo relativo de f(x)

Observando la tabla, el mínimo relativo de f(x) es - 2.

Step2: Encontrar el mínimo relativo de g(x)=(x - 3)^2(x - 6)

Expansión: g(x)=(x^2 - 6x + 9)(x - 6)=x^3-6x^2-6x^2 + 36x+9x - 54=x^3-12x^2 + 45x - 54. Calculamos la derivada: g'(x)=3x^2-24x + 45. Igualamos a cero: 3x^2-24x + 45 = 0. Dividimos entre 3: x^2-8x + 15 = 0. Factorizamos: (x - 3)(x - 5)=0. Las raíces son x = 3 y x = 5. Calculamos la segunda - derivada: g''(x)=6x-24. g''(3)=6(3)-24=-6<0 (máximo relativo). g''(5)=6(5)-24 = 6>0 (mínimo relativo). Evaluamos g(5): g(5)=(5 - 3)^2(5 - 6)=4×(-1)=-4.

Step3: Calcular la diferencia

Diferencia = - 2-(-4)=-2 + 4 = 2.

Answer:

A. 2 units