Question

Complex Analysis

2. (a) (10 marks) Find the least squares approximation of f(x) = x² + 3 over the interval [0, 1] by a function of the form y = aet + bx, where a, b e R. You should write the coefficients a, b as decimal approximations, rounded to two decimal places.

(b) (10 marks) Let g(x) be the least squares approximation you found in the previous problem. So g(x) = ae² + bxr for some scalars a, b. Find the least squares approximation of g(x) over the interval [0, 1] by a function of the form cr² +d.You should write the coefficients c, d as decimal approximations, rounded to two decimal places.

(c) (5 marks) Let h(x) be the function you found in part (b), and g(x) the function you found in part (a). Without actually doing any integral computations,decide which of the following facts must be true:

\text { i. } \int_{0}^{1}\left(g(x)-\left(x^{2}+3\right)\right)^{2} d x \geq \int_{0}^{1}(g(x)-h(x))^{2} d x

\text { ii. } \int_{0}^{1}\left(g(x)-\left(x^{2}+3\right)\right)^{2} d x \leq \int_{0}^{1}(g(x)-h(x))^{2} d x