1. Prove that if f is a continuous function on a closed bounded interval Iand if f(x) is never 0 for x ∈ I, then there is a number m > 0 such thatf(x) ≥ m for all x ∈ I or f(x) ≤ -m for all x ∈ I.

2. Prove that if f is a continuous function on a closed bounded interval [a, b]and if (x0, y0) is any point in the plane, then there is a closest point to(x0, y0) on the graph of f.

3. Find an example of a function which is continuous on a bounded (but not closed) interval I, but is not bounded. Then find an example of a function which is continuous and bounded on a bounded interval I, but does not have a maximum value.

4. Find an example of a function which is continuous on a closed (but not bounded) interval I, but is not bounded. Then find an example of a function which is continuous and bounded on a closed interval I, but does not have a maximum value.

5. Show that if f and g are continuous functions on the interval [a, b] suchthat f(a) g(a) and g(b) f(b), then there is a number c ∈ (a, b) suchthat f(c) = g(c).

6. Use the intermediate value theorem to prove that, if n is a natural number,then every positive number a has a positive nth root.

7. Prove that a polynomial of odd degree has at least one real root.

8. Use the intermediate value theorem to prove that if f is a continuousfunction on an interval [a, b] and if f(x) ≤ m for every x ∈ [a, b), thenf(b) ≤ m.