6. Least squares data fitting [3pt]. We believe that a real number Y is approximately determined by X with the model function
\(Y = a \exp(X) + bX^2 + cX + d.\)
We are given the following table of data for the values of X and Y:
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
X & 0.0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.0 & 2.5 \\
\hline
Y & 0.0 & 0.20 & 0.27 & 0.30 & 0.32 & 0.35 & 0.27 \\
\hline
\end{tabular}
Using the above data points, write down 7 equations in the four unknowns a, b, c, d. The least squares solution to this system is the best fit function. Write down the normal equations for this system, and solve them with MATLAB/Python/Julia. Plot the data points (X, Y) as points and the best fit function.
