let's say we have a function $f \in C^{\infty}[a,b]$ such that $f,f',f'' > 0 \forall x \in [a,b]$
What i would like to prove is that the solution of this boundary problem
$$\left\{ \begin{array}{l} \beta_2 - \beta_0 = 2hf'(a); \\ \beta_i + 4 \beta_{i+1} + \beta_{i+2} = 6f(x_i) \;\;\;\; i = 0...n\\ \beta_{n+2} - \beta_n = 2hf'(b) \end{array} \right. $$
has the following properties (if it is true)
$$ \begin{array}{l} \beta_i > 0 \;\; \forall i \\ \Delta \beta_i > 0 \;\; \forall i \\ \Delta^2 \beta_i > 0 \;\; \forall i \end{array} $$
the $x_i$ are such that $x_{i+1} - x_i = h$.
By the way... the question is i tried to prove by myself... i tried to exploit definition of derivability and convexity, the only thing i got was that the equation in the middle is positive, the finite difference of the equation in the middle is also positive and the second differences too).