analyzing the intervals of a function\npredict which statements are true about the intervals of the…

analyzing the intervals of a function\npredict which statements are true about the intervals of the continuous function. check all that apply.\n$f(x)>0$ over the interval $(-\\infty,3)$.\n$f(x)\\leq0$ over the interval $0,2$.\n$f(x)<0$ over the interval $(-1,1)$.\n$f(x)>0$ over the interval $(-2,0)$.\n$f(x)\\geq0$ over the interval $2,\\infty)$.

analyzing the intervals of a function\npredict which statements are true about the intervals of the continuous function. check all that apply.\n$f(x)>0$ over the interval $(-\\infty,3)$.\n$f(x)\\leq0$ over the interval $0,2$.\n$f(x)<0$ over the interval $(-1,1)$.\n$f(x)>0$ over the interval $(-2,0)$.\n$f(x)\\geq0$ over the interval $2,\\infty)$.

Answer

Explanation:

Step1: Analyze $f(x)>0$ over $(-\infty,3)$

From the table, when $x = - 3,f(-3)=-15<0$, so $f(x)>0$ over $(-\infty,3)$ is false.

Step2: Analyze $f(x)\leq0$ over $[0,2]$

When $x = 0,f(0)=0$; when $x = 1,f(1)=-3<0$; when $x = 2,f(2)=0$. So $f(x)\leq0$ over $[0,2]$ is true.

Step3: Analyze $f(x)<0$ over $(-1,1)$

When $x=-1,f(-1) = 3>0$, so $f(x)<0$ over $(-1,1)$ is false.

Step4: Analyze $f(x)>0$ over $(-2,0)$

When $x=-1,f(-1)=3>0$ and when $x = 0,f(0)=0$, so $f(x)>0$ over $(-2,0)$ is true.

Step5: Analyze $f(x)\geq0$ over $[2,\infty)$

When $x = 2,f(2)=0$ and when $x = 3,f(3)=15>0$. Since the function is continuous, $f(x)\geq0$ over $[2,\infty)$ is true.

Answer:

$f(x)\leq0$ over the interval $[0,2]$, $f(x)>0$ over the interval $(-2,0)$, $f(x)\geq0$ over the interval $[2,\infty)$