Answer
The questions are answered in detail in the Work Step by Step below.
Work Step by Step
1) A function $f(x)$ having the Intermediate Value Property on a closed interval $[a,b]$ means that
- First, it is continuous on $[a,b]$. The graph of the function on this interval is a single, connected line, without any breaks or jumps.
- If $y_0$ is any value between $f(a)$ and $f(b)$, then the horizontal line $y=y_0$ must cross the graph of $f(x)$ at least once, or there must be a value of $c\in[a,b]$ with which $f(c)=y_0$
2) Conditions for Intermediate Value Property on the interval $[a,b]$:
The function $f(x)$ must be continuous on $[a,b]$. Any discontinuity at any points in $[a,b]$ could make the property fail.
3) Consequences for Root Finding $f(x)=0$:
If we can find an interval $[a,b]$ and prove that:
- $f(x)$ is continuous on $[a,b]$
- The values of $f(a)$ and $f(b)$ have different signs.
then we can say that there has been a change of signs from $f(a)$ to $f(b)$ and, by the Intermediate Value Theorem, there must be at least one value of $x=c\in[a,b]$ that $f(c)=0$. In other words, $f(x)=0$ has at least one solution in $[a,b]$.
The graph of $f(x)$ must cross the $x$-axis at least once on the interval $[a,b]$.
