Question

for the function below, find a) the critical numbers; b) the open intervals where the function is increasing; and c) the open intervals where it is decreasing.
f(x) = 12x³ - 81x² - 1008x + 4

a) find the critical number(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
○ a. the critical number(s) is/are .
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
○ b. there are no critical numbers.

Explanation:

Step1: Find the derivative of \( f(x) \)

To find critical numbers, we first find the derivative of \( f(x) = 12x^3 - 81x^2 - 1008x + 4 \). Using the power rule, the derivative \( f'(x) \) is:
\( f'(x) = 36x^2 - 162x - 1008 \)

Step2: Simplify the derivative

We can factor out a common factor of 18 from the derivative:
\( f'(x) = 18(2x^2 - 9x - 56) \)
Now, factor the quadratic \( 2x^2 - 9x - 56 \). We need two numbers that multiply to \( 2 \times (-56) = -112 \) and add to -9. The numbers are -16 and 7. So,
\( 2x^2 - 9x - 56 = 2x^2 - 16x + 7x - 56 = 2x(x - 8) + 7(x - 8) = (2x + 7)(x - 8) \)
Thus, \( f'(x) = 18(2x + 7)(x - 8) \)

Step3: Find critical numbers

Critical numbers occur where \( f'(x) = 0 \) or \( f'(x) \) is undefined. Since \( f'(x) \) is a polynomial, it is defined everywhere. Set \( f'(x) = 0 \):
\( 18(2x + 7)(x - 8) = 0 \)
Since 18 ≠ 0, we solve \( 2x + 7 = 0 \) and \( x - 8 = 0 \).
For \( 2x + 7 = 0 \), we get \( x = -\frac{7}{2} \)
For \( x - 8 = 0 \), we get \( x = 8 \)

Answer:

A. The critical number(s) is/are \( -\frac{7}{2}, 8 \)