5. (10 points) Recall from calculus that given some function g(x), the a you get from solving = 0 is called a critical point of g - this means it could be a minimizer or a "maximizer for g. In this question, we will explore some basic properties and build some intuition on why, for certain loss functions such as the MSE loss, the critical point of the loss will always be the minimizer of the loss. Given some linear model f(x) = yr for some real scalar y, we can write the the mean squared error (MSE) loss of the model f given the observed data {z, y, } { = 1, (a) (1 point) Let's break the loss function above into individual terms. Complete the following sentence by filling in the blanks using one of the options in the parenthesis following each of the blanks: The MSE loss function can be viewed as a sum of n (linear/ quadratic/log- arithmic/exponential) terms, each of which can be treated as a function of ... (b) (3 points) Let's investigate one of the n functions in the summation in the MSE loss function. Define g.(3) = = (v. =))" for: = 1. .n. Recall from calculus that we can use the 2nd derivative of a function to describe its curvature about a certain point ( if it is facing concave up, down. or possibly a point of inflection). You can take the following as a fact: A function is convex if and only if the function = 2nd derivative is non-negative on its domain. Based on this property, verify that g, is a convex function. (c) (2 points) Briefly explain intuitively in words why given a convex function a(z), the critical points we get by solving 32 - 0 minimizes g. You can assume that is a function of a (and not a constant) (d) (3 points) Now that we have shown that each term in the summation of MSE is a convex function, one might wonder if the entire summation is convex given it's a sum of convex functions. While the answer to this for a multivariable function is out of scope for this course, we can still build some intuitions by focusing on single variable functions, (2 points) Let's look at the formal definition of convex functions. Algebraically speaking, s function g(x) is convex if for any two points (21, 9(21))