Question

for the quadratic function f(x)=x² + 2x, answer parts (a) through (f). correct choice below and fill in the answer box(es) within your choice. (type your answer in interval notation. use integers or fractions for any numbers in the expression.) a. f(x)>0 on , and f(x)<0 on b. f(x)>0 on , and f(x) is never negative c. f(x)<0 on , and f(x) is never positive

Explanation:

Step1: Find the roots of the function

Set $f(x)=x^{2}+2x = 0$. Factor out $x$: $x(x + 2)=0$. The roots are $x=0$ and $x=-2$.

Step2: Analyze the sign of the function in intervals

The intervals are $(-\infty,-2)$, $(-2,0)$ and $(0,\infty)$.
For $x<-2$, let $x=-3$, then $f(-3)=(-3)^{2}+2\times(-3)=9 - 6=3>0$.
For $-2For $x>0$, let $x = 1$, then $f(1)=1^{2}+2\times1=1 + 2=3>0$.

Answer:

A. $f(x)>0$ on $(-\infty,-2)\cup(0,\infty)$, and $f(x)<0$ on $(-2,0)$