Determine a cubic spline f : [0, 3] R such that

f(xi) = yi , for i = 1, 2, 3, (x1, x2, x3) = (0, 2, 3), (y1, y2, y3) = (0, 1, 2);

f _[xi,xi+1] is polynomial of degree less or equal to three for i = 1, 2;

f, f ' , f '' are continuous in (0, 3);

f ' (0⁺) = 0, f ' (3^-) = 0.

This can be done constructing a system of linear equations and solving it in Matlab. Plot your solution and compare it to the output of the Matlab function spline:

x = [0 2 3];

y = [0 0 1 2 0];

cs = spline(x, y);

plot(linspace(0,3),ppval(cs,linspace(0,3)),-r);