NUn dann setzen wir
\(\small f_o(x) \, := \, a \; k^{x} \)
und setzen ein fo(X(j))=Y(j), X={1,2,3,4,5,6,7,8,9}
\(\small Fxy \, := \, \left\{ a \; k = 8, a \; k^{2} = \frac{89}{10}, a \; k^{3} = \frac{48}{5}, a \; k^{4} = \frac{53}{5}, a \; k^{5} = \frac{117}{10}, a \; k^{6} = 13, a \; k^{7} = \frac{71}{5}, a \; k^{8} = \frac{78}{5}, a \; k^{9} = \frac{171}{10} \right\} \)
\(\small \sum_{j=1}^{9}\left(f_o\left(X\left(j \right) \right) - Y\left(j \right) \right)^{2} \)
\(\small =Q \, := \, a^{2} \; k^{18} + a^{2} \; k^{16} + a^{2} \; k^{14} + a^{2} \; k^{12} + a^{2} \; k^{10} + a^{2} \; k^{8} + a^{2} \; k^{6} + a^{2} \; k^{4} + a^{2} \; k^{2} - 34.2 \; a \; k^{9} - 31.2 \; a \; k^{8} - 28.4 \; a \; k^{7} - 26 \; a \; k^{6} - 23.4 \; a \; k^{5} - 21.2 \; a \; k^{4} - 19.2 \; a \; k^{3} - 17.8 \; a \; k^{2} - 16 \; a \; k + 1391.03 \)
dQ:={Derivative(Q,a ),Derivative(Q,k)}
\(\small dQ \, := \, \left\{ 2 \; a \; k^{18} + 2 \; a \; k^{16} + 2 \; a \; k^{14} + 2 \; a \; k^{12} + 2 \; a \; k^{10} - \frac{171}{5} \; k^{9} + 2 \; a \; k^{8} - \frac{156}{5} \; k^{8} - \frac{142}{5} \; k^{7} + 2 \; a \; k^{6} - 26 \; k^{6} - \frac{117}{5} \; k^{5} + 2 \; a \; k^{4} - \frac{106}{5} \; k^{4} - \frac{96}{5} \; k^{3} + 2 \; a \; k^{2} - \frac{89}{5} \; k^{2} - 16 \; k,\\ 18 \; a^{2} \; k^{17} + 16 \; a^{2} \; k^{15} + 14 \; a^{2} \; k^{13} + 12 \; a^{2} \; k^{11} + 10 \; a^{2} \; k^{9} - \frac{1539}{5} \; a \; k^{8} + 8 \; a^{2} \; k^{7} - \frac{1248}{5} \; a \; k^{7} - \frac{994}{5} \; a \; k^{6} + 6 \; a^{2} \; k^{5} - 156 \; a \; k^{5} - 117 \; a \; k^{4} + 4 \; a^{2} \; k^{3} - \frac{424}{5} \; a \; k^{3} - \frac{288}{5} \; a \; k^{2} + 2 \; a^{2} \; k - \frac{178}{5} \; a \; k - 16 \; a \right\} \)
Solve(dQ,{a,k})
\(\small \left\{ \left\{ a = 0, k = 0 \right\} , \left\{ a = 7.28716, k = 1.09973 \right\} , \left\{ a = -2.55061, k = -0.79338 \right\} \right\} \)
\(\small \)
