Prove that `e^(x) ge 1 +x` and hence `e^(x) +sqrt(1+e^(2x))ge(1+x)+sqrt(2+2x+x^(2)) forall` x in R

Explanation

To prove that \( e^x \geq 1 + x \) and hence \( e^x + \sqrt{1 + e^{2x}} \geq 1 + x + \sqrt{2 + 2x + x^2} \) for all \( x \in \mathbb{R} \), we will follow these steps:

### Step 1: Prove \( e^x \geq 1 + x \)

1. **Consider the function** \( f(x) = e^x - (1 + x) \).
2. **Calculate the derivative**:
\[
f'(x) = e^x - 1
...