Question

find the intervals on which ( f(x) ) is increasing, the intervals on which ( f(x) ) is decreasing and the local extrema.
( f(x)=-2 x^{2}-8 x - 21 )
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function is increasing on ( (-infty,-2) ).
(type your answer in interval notation. type integers or simplified fractions. 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 in interval notation. type integers or simplified fractions. use a comma to separate answers as needed.)
b. the function is never decreasing.

Explanation:

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

The derivative of \( f(x)=-2x^{2}-8x - 21 \) is \( f^\prime(x)=-4x - 8 \).

Step2: Find the critical point

Set \( f^\prime(x)=0 \), so \( -4x - 8 = 0 \). Solving for \( x \):

$$
LATEXBLOCK0
$$

Step3: Determine the intervals of increase and decrease

• For \( x < - 2 \), let \( x=-3 \). Then \( f^\prime(-3)=-4\times(-3)-8 = 12 - 8=4>0 \). So \( f(x) \) is increasing on \( (-\infty,-2) \).
• For \( x > - 2 \), let \( x = 0 \). Then \( f^\prime(0)=-4\times0 - 8=-8<0 \). So \( f(x) \) is decreasing on \( (-2,\infty) \).

Answer:

The function is decreasing on \( (-2,\infty) \).