Consider the Hyperbolic Function below.

\[ P(x,\lambda,\tau) = x\lambda + \sqrt{(x\lambda)^2 + \tau^2}. \]

How many of the following statements are true:

- \(P(x,\lambda,\tau) \in C^\infty\)
- \(P(x,\lambda,\tau)\) is asymptotically tangent to the straight lines \(r_1(x)=2\lambda x\) and \(r_2(x)=0\) for \(\tau > 0\)
- \(P(x,\lambda,\tau) \geq 2\lambda x \, \forall x, \lambda \geq 0, \, \tau \geq 0\)
- \(P(0,\lambda,\tau) = \tau, \, \lambda \geq 0, \, \tau \geq 0\)
- \(P(0,\lambda,\tau)\) is a convex increasing function of \(x\) for \(\lambda \geq 0, \, \tau \geq 0\)

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