## University Calculus: Early Transcendentals (3rd Edition)

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 $[a,b]$.