To determine the absolute maximum and minimum values of a function, we need to follow these steps:
Step 1: Identify the derivative of the function
The derivative of the function f(x) = x³ - 6x² - 63x + 2 is f'(x) = 3x² - 12x - 63.
Step 2: Find the critical points of the function
To find the critical points of the function, we need to set the derivative equal to zero:
0 = 3x² - 12x - 63
By solving the above quadratic equation, three potential critical points are obtained, which are -1, -3, and 14. But only the critical point that falls within the interval (-4, 0), which is -3, is taken into account.
Step 3: Evaluate the function at each critical point and at the endpoints of the interval
The function will be evaluated at the value of the critical point (-3) as well as the lower limit (-4) and upper limit (0) of the interval. Following are the evaluated values:
f(-4) = 94,
f(-3) = 110,
f(0) = 2.
Step 4: Determining the maximum and minimum value
From the values calculated, we see that 110 is our highest value and 2 our lowest. Hence, 110 is the absolute maximum and 2 is the absolute minimum over the interval (-4, 0). So, the answer is:
Critical points: {-3},
Function values at critical points and endpoints: {110, 94, 2},
Absolute Maximum: 110,
Absolute Minimum: 2.