Please do question 3, thx

The solution P_n (x) to the Least Squares Approximation problem of f by a polynomial of degree at most n is given explicitly in terms of orthogonal_0(x),_1 (x)-1 ..,_n (x), where_2 is a polynomial of degree j, by P_n(x) = sigma_j = 0^n a_j_3 (x), a_3 = (f,_j)/(_j,_j). (a) Let p-n be the space of polynomials of degree at most n. Prove that the error f - P_n is orthogonal to this space, i.e (f - P_n, q) = 0 for any q P_n. (b) Using the analogy of vectors interpret this result geometrically (recall the concept of orthogonal projection). (a) Obtain the first 4 Legendre polynomials in [-1, 1]. (b) Find the least squares polynomial approximations of degrees 1, 2, and 3 for the function f(x) = e^x on [-1, 1]. (c) What is the polynomial least squares approximation of degree 4 for f(x) = x^3 on [-1, 1]? Explain. Plot the monic Chebyshev polynomials T_0 (x), T_2(x), (T_3(x), and T_4 (x)