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 (-∞, 3).\n f(x)≤0 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)≥0 over the interval 2, ∞).\n x f(x)\n-3 -15\n-2 0\n-1 3\n0 0\n1 -3\n2 0\n3 15

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 (-∞, 3).\n f(x)≤0 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)≥0 over the interval 2, ∞).\n x f(x)\n-3 -15\n-2 0\n-1 3\n0 0\n1 -3\n2 0\n3 15

Answer

Explanation:

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

At $x = - 3$, $f(-3)=-15<0$, so this statement is false.

Step2: Check $f(x)\leq0$ 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 $[0,2]$, this statement is true.

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

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

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

When $x=-2$, $f(-2)=0$ and when $x = 0$, $f(0)=0$, and for values between $-2$ and $0$ we don't have more - detailed information, but since the function is continuous and $f(-2)=f(0)=0$, $f(x)$ is not strictly greater than $0$ over $(-2,0)$, so this statement is false.

Step5: Check $f(x)\geq0$ 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 $[2,\infty)$, this statement is true.

Answer:

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