Question

1. if n(x) is a continuous function, how many roots are guaranteed by the intermediate value theorem on (0,12)?

Explanation:

Step1: Recall Intermediate - Value Theorem

If a function $y = n(x)$ is continuous on the closed - interval $[a,b]$ and $k$ is a number between $n(a)$ and $n(b)$, then there exists at least one number $c$ in the open interval $(a,b)$ such that $n(c)=k$. For a root, $k = 0$.

Step2: Check sign changes

We have the following pairs of values:
When $x = 0$, $n(0)=5$; when $x = 1$, $n(1)=-4$. Since $n(0)=5>0$ and $n(1)=-4 < 0$, by the Intermediate - Value Theorem, there is at least one root in the interval $(0,1)$.
When $x = 4$, $n(4)=2$; when $x = 7$, $n(7)=-1$. Since $n(4)=2>0$ and $n(7)=-1 < 0$, by the Intermediate - Value Theorem, there is at least one root in the interval $(4,7)$.
When $x = 10$, $n(10)=4$; when $x = 12$, $n(12)=-5$. Since $n(10)=4>0$ and $n(12)=-5 < 0$, by the Intermediate - Value Theorem, there is at least one root in the interval $(10,12)$.

Answer:

3