(Solved): 3. A clamped cubic spline \( S \) for a function \( f \) is defined by \[ s(x)=\left\{\begin{array ...

3. A clamped cubic spline \( S \) for a function \( f \) is defined by \[ s(x)=\left\{\begin{array}{ll} s_{0}(x)=1+B x+2 x^{2}-2 x^{3}, & \text { if } 0 \leqslant x<1 \\ s_{1}(x)=1+b(x-1)-4(x-1)^{2}+7(x-1)^{3}, & \text { if } 1 \leqslant x \leqslant 2 \end{array}\right. \] Find (i) \( B \) and \( b \) and then determine \( f^{\prime}(0) \) and \( f^{\prime}(2) \) via the clamped boundary conditions. (Do not use code.)