# Solutions to Problem Set 8

Math 310-01/02

9-22-2017

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 .

is continuous,

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

Therefore,

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.

To think is not enough; you must think of something. - Jules Renard

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