Question

determine the open intervals on which the function is increasing, decreasing, or constant. (enter your answers using interval notation. if an answer does not exist, enter dne.)
$f(x)=x^{2}-12x$

Explanation:

Step1: Find the derivative of the function

The function is \( f(x)=x^{2}-12x \).
Using the power rule \( (x^{n})^\prime = nx^{n - 1} \), the derivative \( f^\prime(x)=2x-12 \).

Step2: Find the critical point

Set \( f^\prime(x) = 0 \), so \( 2x-12=0 \).
Solving for \( x \):
\( 2x=12 \), \( x = 6 \).

Step3: Determine the sign of the derivative in intervals

• For the interval \( (-\infty,6) \), let's take a test - point \( x = 5 \). Then \( f^\prime(5)=2\times5 - 12=10 - 12=-2<0 \).
• For the interval \( (6,\infty) \), let's take a test - point \( x = 7 \). Then \( f^\prime(7)=2\times7 - 12=14 - 12 = 2>0 \).

Answer:

• Increasing: \( (6,\infty) \)
• Decreasing: \( (-\infty,6) \)
• Constant: DNE