Question

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.

( f(x)=7 + 4x - x^{2} )

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the function is increasing on
(type your answer using interval notation. use a comma to separate answers as needed.)

b. the function is never increasing.

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 \(x\):

$$
LATEXBLOCK0
$$

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 \((-\infty,2)\)