Question

find any relative extrema of the function. list each extremum along with the x - value at which it occurs. identify intervals over which the function is increasing and over which it is decreasing. then sketch a graph.
$f(x)=-5 - 3x - x^{2}$
describe any relative extrema. select the correct choice below and, if necessary, fill in the answer box(es) to within your choice.
a. the relative minimum point(s) is/are and the relative maximum point(s) is/are
(simplify your answers. type ordered - pairs, using integers or fractions. use a comma to separate answers as needed.)
b. the relative maximum point(s) is/are and there are no relative minimum points.
(simplify your answer. type an ordered - pair, using integers or fractions. use a comma to separate answers as needed.)
c. the relative minimum point(s) is/are and there are no relative maximum points.
(simplify your answer. type an ordered - pair, using integers or fractions. use a comma to separate answers as needed.)
d. there are no relative minimum points and there are no relative maximum points.
identify any intervals over which the function is increasing or decreasing. select the correct choice below and fill in the answer box(es) within your choice.
a. the function $f(x)$ is increasing over the interval(s) and decreasing over the interval(s)
(type your answers in interval notation. use integers or fractions for any numbers in the expressions. use a comma to separate answers as needed.)
b. the function $f(x)$ is increasing over the interval(s) and is not decreasing anywhere.
(type your answer in interval notation. use integers or fractions for any numbers in the expressions. use a comma to separate answers as needed.)
c. the function $f(x)$ is decreasing over the interval(s) and is not increasing anywhere.
(type your answer in interval notation. use integers or fractions for any numbers in the expressions. use a comma to separate answers as needed.)

Explanation:

Step1: Find the derivative of the function

Given $f(x)=-5 - 3x - x^{2}$, using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we have $f^\prime(x)=-3-2x$.

Step2: Find the critical points

Set $f^\prime(x) = 0$. So, $-3-2x = 0$. Solving for $x$ gives $x=-\frac{3}{2}$.

Step3: Determine if the critical point is a relative extrema

We use the second - derivative test. Find the second - derivative $f^{\prime\prime}(x)=\frac{d}{dx}(-3 - 2x)=-2$. Since $f^{\prime\prime}(x)=-2<0$, the function has a relative maximum at $x =-\frac{3}{2}$.
Substitute $x =-\frac{3}{2}$ into the original function $f(-\frac{3}{2})=-5-3\times(-\frac{3}{2})-(-\frac{3}{2})^{2}=-5+\frac{9}{2}-\frac{9}{4}=-\frac{20}{4}+\frac{18}{4}-\frac{9}{4}=-\frac{11}{4}$. So the relative maximum point is $(-\frac{3}{2},-\frac{11}{4})$.

Step4: Determine the intervals of increase and decrease

We consider the sign of $f^\prime(x)$.
Let's test intervals. Choose a value less than $-\frac{3}{2}$, say $x=-2$. Then $f^\prime(-2)=-3-2\times(-2)=-3 + 4 = 1>0$, so the function is increasing on the interval $(-\infty,-\frac{3}{2})$.
Choose a value greater than $-\frac{3}{2}$, say $x=-1$. Then $f^\prime(-1)=-3-2\times(-1)=-3 + 2=-1<0$, so the function is decreasing on the interval $(-\frac{3}{2},\infty)$.

Answer:

For the relative extrema:
B. The relative maximum point(s) is $(-\frac{3}{2},-\frac{11}{4})$ and there are no relative minimum points.
For the intervals of increase and decrease:
A. The function $f(x)$ is increasing over the interval $(-\infty,-\frac{3}{2})$ and decreasing over the interval $(-\frac{3}{2},\infty)$.