find the intervals on which ( f(x) ) is increasing and the intervals on which ( f(x) ) is decreasing. then…

find the intervals on which ( f(x) ) is increasing and the intervals on which ( f(x) ) is decreasing. then sketch the graph. add horizontal tangent lines.\n\n( f(x)=7 + 4x - x^{2} )\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the function is increasing on\n(type your answer using interval notation. use a comma to separate answers as needed.)\n\nb. the function is never increasing.

find the intervals on which ( f(x) ) is increasing and the intervals on which ( f(x) ) is decreasing. then sketch the graph. add horizontal tangent lines.\n\n( f(x)=7 + 4x - x^{2} )\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the function is increasing on\n(type your answer using interval notation. use a comma to separate answers as needed.)\n\nb. the function is never increasing.

Answer

Explanation:

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

The function is (f(x)=7 + 4x-x^{2}). Using the power rule ((x^{n})^\prime=nx^{n - 1}), the derivative (f^\prime(x)=(7)^\prime+(4x)^\prime-(x^{2})^\prime). Since ((c)^\prime = 0) (where (c) is a constant) and ((ax^{n})^\prime=anx^{n-1}), we have (f^\prime(x)=4-2x).

Step2: Find the critical points

Set (f^\prime(x) = 0). So, (4-2x=0). Solving for (x): [ \begin{align*} 4-2x&=0\ -2x&=-4\ x& = 2 \end{align*} ]

Step3: Test the intervals

We have two intervals to test: ((-\infty,2)) and ((2,\infty)).

  • For the interval ((-\infty,2)), let's choose (x = 0). Then (f^\prime(0)=4-2\times0=4>0).
  • For the interval ((2,\infty)), let's choose (x = 3). Then (f^\prime(3)=4-2\times3=-2<0).

Answer:

The function (f(x)=7 + 4x - x^{2}) is increasing on ((-\infty,2)) and decreasing on ((2,\infty)). So the answer is A. The function is increasing on ((-\infty,2))