Functions

Step 1:
Find the points where the functions intersect:


So, the critical points are:

Step 2:
Determine the function in different intervals:

For :
For :
For :

Hence, the function can be written as:

Step 3:
Check differentiability at :

Left-hand derivative (LHD): Right-hand derivative (RHD):
Since,
Therefore, the function is not differentiable at .

Final Conclusion:

To solve the given problem, we start by analyzing the expression for :

This expression represents a complex number in polar form. According to Euler's formula, we can rewrite this as:

We are asked to find the product:

Substituting the expression for each , we have:

Thus, the infinite product becomes:

Using properties of exponents, we can combine these into a single exponential:

This is an infinite geometric series with first term and common ratio . The sum of an infinite geometric series is given by:

Substituting the values, we find:

Therefore, the exponent of becomes , giving us the complete expression:

Now, evaluating this, we use the Euler's formula:

Calculating these values, and , we find:

Therefore, the product of the series is .