Show that `1+xin(x+sqrt(x^2+1))geqsqrt(1+x^2)` for all `xgeq0.`

Explanation

To prove that \( 1 + x \ln(x + \sqrt{x^2 + 1}) \geq \sqrt{1 + x^2} \) for all \( x \geq 0 \), we will define a function and analyze its behavior.

### Step 1: Define the function
Let
\[ f(x) = 1 + x \ln(x + \sqrt{x^2 + 1}) - \sqrt{1 + x^2} \]
We need to show that \( f(x) \geq 0 \) for all \( x \geq 0 \).
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