1. Prove that there is a positive integer n such that is not prime.
There are many possibilities. For example, if , then
But 15 is not prime.
2. Suppose that f is a continuous function,
Prove that has a root between and .
By the Intermediate Value Theorem, for some c between 3 and 6 --- that is, has a root between and .
3. Suppose that f is a differentiable function which satisfies
Prove that .
Apply the Mean Value Theorem to f on the interval . The theorem says that there is a number c such that and
4. Let .
(a) Use the Intermediate Value Theorem to prove that has at least one root.
(b) Use Rolle's theorem to prove that f has only one root.
(a) f is continuous.
Since and , the Intermediate Value Theorem implies that f has at least one root between 1 and -1.
(b) Suppose f has two roots a and b with , so . By Rolle's Theorem, f has a critical point c between a and b. However,
Hence, f has no critical points. This contradiction shows that f can't have two roots. Since by (a) it has at least one root, it must have only one root.
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