Is the year 2050. A new pandemic has spread across the world, and a group of students are selected to find the solution to the pandemic by solving a linear regression that will provide an estimate number of the right amount of money to be invested in the vaccine developers: (i) more than the "optimal" investment will have a heavy load on the taxpayer and will risk the lives of the poorer, (ii) less than the "optimal" investment will imply a delay in the vaccine production and by consequence risk the lives of the world population. The "optimal" investment is found by producing a linear regression model over 10 sample observations of the dependent variable (the investment to be made) Y (Y, Y2, ..., Y10), with the independent variable being X (1,2,..., 10). Part I: Differential Calculus, Integral Calculus, Optimisation and Linear Algebra You can observe only two samples, the last two, Yg = 3, and Y10 = 2. The first eight sample observations of the dependent variable (Y, Y2,..., Ys) are derived from the following rules (If you can't calculate a value substitute Y = 0 for i the value that you can't calculate): a. (5 pts) Find the first observation (Y) of the samples to be regressed, by calculating the second derivative of Y 1 5x + 6x + 2 evaluated at x = 1. b. (10 pts) Find the second observation (Y) of the samples to be regressed, by calculating the integral of y = (x + 1) 2x S evaluated at x = 2. Consider the constant of integration C=0. -d.x c. (10 pts) Find the third and fourth (Y3, Y4) observation of the samples to be regressed, by calculating the stationary points (The values of and ) of y = (4x + 1)(x - 1)