Let x1, . . . , xn be real numbers representing positions on a number line. Let w1, . . . , wn be positive real numbers representing the importance of each of these positions. Consider the quadratic function: f(θ) = 1 2 Pn i=1 wi(θ − xi) 2 . What value of θ minimizes f(θ)? Show that the optimum you find is indeed a minimum. What problematic issues could arise if some of the wi 's are negative? [NOTE: You can think about this problem as trying to find the point θ that's not too far away from the xi 's. Over time, hopefully you'll appreciate how nice quadratic functions are to minimize.] [HINT: View f(θ) as a quadratic function in θ, i. E. F(θ) = αθ2 +βθ +γ, where α, β, γ are real numbers depending on wi 's and xi 's.]