A function is convex if and only if the function's 2nd derivative is non-negative on its domain. Solve for the question in the photo.

(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 singlevariable functions. i. (2 points) Let's look at the formal denition of convex functions. Algebraically speaking, a function 9(a) is convex if for any two points (9:1, 9(a)) and (292, 9(322 on the function: 9(6331 + (1 - (3)332) S cg($1)+(1- C)9($2) Homework #7 8 for any real constant 0 S c S 1. Intuitively, the above denition says that, given the plot of a convex function 9(a), if you connect 2 randomly chosen points on the function, the line segment will always lie on or above 9(a) (try this with the graph of y = :32). Using this denition, show that if 9(29) and Mm) are both convex functions, their sum 9(a) + Mac) will also be a convex function. ii. (1 point) Based on what you have shown in the previous part, explain intu itively why the sum of n convex functions is still a convex function when n > 2. (e) (1 point) Finally, explain why in our case that, when we solve for the critical point of the MSE loss function by taking the gradient with respect to the parameter and setting the expression to 0, it is guranteed that the solution we nd will minimize the MSE loss