stability of cubic spline

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Let f(x) be a cubic polynomial. Suppose we interpolate f(x) by a cubic spline. That is, suppose that x0,x1,…,xn are a set of x-values, and that we make a cubic spline through the points (x0,f(x0)),(x1,f(x1)),…,(xn,f(xn)).a. Suppose that we make a clamped cubic spline, with derivative at x0 given by f′(x0) and derivative at xn given by f′(xn). Show that the cubic spline we get is f(x) itself.b. Suppose that we make a natural cubic spline. Show that the cubic spline we get can’t be f(x) itself.c. Suppose that we make a cubic spline using the “not-a-knot” conditions. Show that the cubic spline we get is f(x) itself.Repeat problem 4, but now let f(x) be a linear polynomial.a. Is the clamped cubic spline f(x) itself? Show that it is or isn’t.b. Is the natural cubic spline f(x) itself? Show that it is or isn’t.c. Is the “not-a-knot” cubic spline f(x) itself? Show that it is or isn’t.

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