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 + 8x - 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 ( (-infty, 4) ).
(type your answer using interval notation. use a comma to separate answers as needed.)

b. the function is never increasing.

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

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

b. the function is never decreasing.

Explanation:

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

Given \(f(x)=7 + 8x-x^{2}\), using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(f^\prime(x)=(7)^\prime+(8x)^\prime-(x^{2})^\prime\).
Since \((c)^\prime = 0\) (where \(c\) is a constant) and \((ax^{n})^\prime=anx^{n-1}\), then \(f^\prime(x)=0 + 8-2x=8 - 2x\).

Step2: Find the critical points

Set \(f^\prime(x)=0\), so \(8 - 2x=0\).
Solving for \(x\):

$$
LATEXBLOCK0
$$

Step3: Determine the sign of \(f^\prime(x)\) in different intervals

• For \(x<4\), let \(x = 3\). Then \(f^\prime(3)=8-2\times3=8 - 6=2>0\). So \(f(x)\) is increasing on \((-\infty,4)\).
• For \(x>4\), let \(x = 5\). Then \(f^\prime(5)=8-2\times5=8 - 10=-2<0\). So \(f(x)\) is decreasing on \((4,\infty)\)

Answer:

The function is decreasing on \((4,\infty)\)