Suppose that f and g are two functions both continuous on the
interval [a, b], and such that (b) = p and f(b) = g(a) = q
where p does not equal to q. Sketch typical graphs of two such functions . Then apply the intermediate value theorem to the function
h(x) = f(x) - g(x) to show that f(c) = g(c) at some point c of
(a, b).
An example case is illustrated by the sin(x) and cos(x) functions between π/2 and 2π (see link at end of post).
"Intermediate value theorem":
"for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value".
Since p≠q, either p>q or q>p. Assume p>q, b>a.
Then (a), and f(b)<g(b), so if
h(x)=f(x)-g(x)
then h(a)>0, and h(b)<0.
You will need to prove, using the intermediate value theorem, that h(x0)=0 at some point a<x0<b.
Proof is similar when p<q.
http://imageshack.us/photo/my-images/821/1316321469.png/
To sketch typical graphs of two functions, f and g, both continuous on the interval [a, b], we need to consider their properties. Since (b) = p and f(b) = g(a) = q, where p ≠ q, we can visually depict these conditions as follows:
1. Start by plotting an interval on the x-axis from a to b.
2. Mark points (a, p) and (b, q) as coordinates on the graph.
3. Draw a smooth curve that connects these points. This represents the graph of function f.
4. Now, draw a parallel curve that connects the points (a, q) and (b, p). This corresponds to the graph of function g.
Keep in mind that these are just general sketches. The specific shape and characteristics of f and g will depend on the functions themselves.
Next, we'll apply the intermediate value theorem to the function h(x) = f(x) - g(x) to show that f(c) = g(c) at some point c in the interval (a, b).
The intermediate value theorem states that if a function h(x) is continuous on a closed interval [a, b] and takes on two different values p and q at the endpoints, then for any value r between p and q, there exists at least one point c in (a, b) where h(c) = r.
In our case, we have h(x) = f(x) - g(x), which is the difference between the two functions.
Step 1: Show that h(x) is continuous.
Since f(x) and g(x) are both continuous on [a, b], their difference, h(x) = f(x) - g(x), will also be continuous on [a, b] as subtracting two continuous functions yields a continuous function.
Step 2: Show that h(a) and h(b) have opposite signs.
We can see that h(a) = (a) = p - q and h(b) = f(b) - g(b) = q - p. Since p ≠ q, the signs of h(a) and h(b) will be opposite.
Step 3: Apply the intermediate value theorem.
According to the intermediate value theorem, since h(x) is continuous on [a, b] and h(a) and h(b) have opposite signs, there must exist at least one point c in the interval (a, b) such that h(c) = 0.
Thus, h(c) = f(c) - g(c) = 0, which implies that f(c) = g(c) at some point c in the interval (a, b).
This demonstrates that there exists at least one point c where the functions f and g intersect, proving f(c) = g(c) for some c in (a, b).