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
(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 derivative of \( f(x)=7 + 8x - x^{2} \) is \( f^{\prime}(x)=8-2x \).

Step2: Set the derivative equal to zero

Set \( f^{\prime}(x) = 0 \), so \( 8-2x=0 \). Solving for \( x \), we get \( 2x = 8 \), then \( x = 4 \).

Step3: Determine the sign of the derivative in intervals

• For \( x<4 \), let \( x = 3 \). Then \( f^{\prime}(3)=8 - 2\times3=8 - 6 = 2>0 \).
• For \( x>4 \), let \( x = 5 \). Then \( f^{\prime}(5)=8 - 2\times5=8 - 10=-2<0 \).

Answer:

A. The function is increasing on \( (-\infty,4) \)