## Differential equation (II)

Let be a differentiable function such that and

Find an explicit formula for .

**Solution**

Let us consider the function which is clearly differentiable. Hence,

Thus,

## Differential equation (I)

Let be a differentiable function such that and

Find an explicit formula for .

**Solution**

We have successively

which satisfies the given conditions.

## An infimum

Let be the vector space of all continuous functions such that . Evaluate

## Maximum value of function

Let . Find the maximum vale of the function

**Solution**

Using the AM – GM inequality we have

Hence,

Equality holds when

Since there exists an such that the equation has at least one root. The minimum follows.

## Infinite product

Prove that