Final answer:
To find the absolute minimum of the function x³ - 9x² + 27x - 3 in the interval [-4, -1], find the critical points and check the endpoints. The absolute minimum value is -18.
Step-by-step explanation:
To find the absolute minimum of the function x³ - 9x² + 27x - 3 in the interval [-4, -1], we can start by finding the critical points of the function. To do this, we take the derivative of the function and set it equal to zero. The derivative of the function is 3x² - 18x + 27. Setting this equal to zero, we get 3(x-3)(x-3) = 0. Therefore, the critical point is x = 3.
Now, we need to check the endpoints of the interval. Plugging in x = -4 and x = -1 into the function, we get f(-4) = 43 and f(-1) = -18.
We must then evaluate the function at the critical points and the endpoints of the interval to find the absolute minimum value on the specified interval.
However, the quadratic equation provided in the student's question does not apply to this function and is likely from a different context.
Comparing the values at the critical point and the endpoints, we find that the absolute minimum value of the function in the interval [-4, -1] is -18.