Hyperbolic function properties

Algebra Level pending

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:

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