CPM Homework Help : APCALC Problem 3-27

If $f(15) = -3$ and $f(20) = 4$ , must $f$ have a root between $x = 15$ and $x = 20$ ? Explain why or why not. Be sure to include sketches that support your reasoning.

Hint:

First quadrant, x axis with 3 unequal tick marks labeled, from left to right, a, c, & b, y axis with 3 equal tick marks, labeled from bottom to top, f of b, m, &, f of a, horizontal line starting at (0, comma m), continuous curve starting at the point (a, comma f of A), turning up below x axis between tick marks, c & b, turning down below horizontal line, ending at the point (b, comma f of b). Intermediate Value Theorem

Let $f$ be a function continuous on the closed interval $[a,b]$ . Then for every value $m$ between $f (a)$ and $f (b)$ there exists a value $c$ , $a < c < b$ , such that $f (c) = m$ .

Hint:

Notice the condition of the Intermediate Value Theorem.