# 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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