determine the open intervals on which the function is increasing, decreasing, or constant. (enter your…

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.)\n$f(x)=x^{2}-12x$\n

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.)\n$f(x)=x^{2}-12x$\n

Answer

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