Question 1.6.7. from Matrix Analysis & Applied Linear Algebra  f[x_] := Sin[Pi*x] f[x] myList[x_, x0_, x1_, x2_, x3_] := x0 + x x1 + x^2*x2 + x^3*x3 myList[x, x0, x1, x2, x3] rExpandedVars[x_, x0_, x1_, x2_, x3_] := Integrate[(f[x] - myList[x, x0, x1, x2, x3])^2, {x, 0, 1}] rExpandedVars[x, x0, x1, x2, x3] dX0[x_, x0_, x1_, x2_, x3_] := D[rExpandedVars[x, x0, x1, x2, x3], x0] dX1[x_, x0_, x1_, x2_, x3_] := D[rExpandedVars[x, x0, x1, x2, x3], x1] dX2[x_, x0_, x1_, x2_, x3_] := D[rExpandedVars[x, x0, x1, x2, x3], x2] dX3[x_, x0_, x1_, x2_, x3_] := D[rExpandedVars[x, x0, x1, x2, x3], x3] dX0[x, x0, x1, x2, x3] dX1[x, x0, x1, x2, x3] dX2[x, x0, x1, x2, x3] dX3[x, x0, x1, x2, x3] Expand[dX0[x, x0, x1, x2, x3]] Expand[dX1[x, x0, x1, x2, x3]] Expand[dX2[x, x0, x1, x2, x3]] Expand[dX3[x, x0, x1, x2, x3]] set the results eual to 0 and it gives the extremum. That we are looking for. |