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**Makall** Let f be defined on [a, b] and let the nodes $\displaystyle a = x_{0} < x_{1} < x_{2} = b$ be given. A quadratic spline interpolating function S consists of the quadratic polynomial $\displaystyle S_{0} = a_{0} + b_{0}(x - x_{0}) + c_{0}(x - x_{0})^2 on [x_{0}, x_{1}]$ and $\displaystyle S_{1} = a_{1} + b_{1}(x - x_{1}) + c_{1}(x - x_{1})^2 on [x_{1}, x_{2}]$ such that

$\displaystyle

(i) S(x_{0}) = f(x_{0}), S(x_{1}) = f(x_{1}), S(x_{2}) = f(x_{2})$

$\displaystyle (ii) S \in C^1[x_{0}, x_{2}]$

Show that conditions (i) and (ii) lead to five equations in the six unknowns $\displaystyle a_{0}, b_{0}, c_{0}, a_{1}, b_{1}, c_{1}$. What additional condition can we impose to make the solution unique?

How do I find the five equations and make it unique?