1. For a graph of Gibbs free energy $G$ vs extent of reaction $\xi$:

   • The reaction proceeds spontaneously in the direction in which $G$ decreases.
   • At equilibrium, $G$ is minimum, so: $\left(\frac{dG}{d\xi}\right)_{T,P}=0$

2. Interpreting the usual $G$ vs extent of reaction curve:

   • Point $(b)$ is the minimum of the curve $\Rightarrow$ equilibrium.
   • Point $(a)$ lies to the left of the minimum. Moving forward in reaction decreases $G$, so reaction is spontaneous.
   • Point $(c)$ lies to the right of the minimum. Further forward progress increases $G$, so in the forward direction reaction is non-spontaneous.

3. Now check each statement:

   A. Reaction is spontaneous at $(a)$ and $(b)$

   • At $(a)$: true
   • At $(b)$: false, because $(b)$ is equilibrium
     So, A is false.

   B. Reaction is at equilibrium at point $(b)$ and non-spontaneous at point $(c)$

   • $(b)$ equilibrium: true
   • $(c)$ non-spontaneous (forward direction): true
     So, B is true.

   C. Reaction is spontaneous at $(a)$ and non-spontaneous at $(c)$

   • $(a)$ spontaneous: true
   • $(c)$ non-spontaneous: true
     So, C is true.

   D. Reaction is non-spontaneous at $(a)$ and $(b)$

   • $(a)$ is spontaneous, so false
   • $(b)$ is equilibrium, not spontaneous
     Hence D is false.

4. Therefore, the true statements are: $\boxed{B \text{ and } C}$

So, the number of true statements is: $\boxed{2}$