Given a differentiable error function that measures the total squared vertical…
MATHEMATICS · Single correct · A Level · Mathematics · Differentiation · Differentiation
Problem
Given a differentiable error function E(p) that measures the total squared vertical deviation between a model that depends on a single parameter p and a set of data points, which statement best justifies why the value of p satisfying E′(p)=0 is a candidate for the optimal fit?
(a) Because E′(p)=0 indicates that E(p) stops decreasing at that point; since E(p)≥0, the first stationary point encountered must be the global minimum.
(b) Because E′(p)=0 forces E(p)=0, eliminating all error between the model and the data and thus proving a perfect fit.
(c) Because E′(p)=0 indicates that infinitesimal changes in p do not change E to first order, so the point is a critical point that could correspond to the desired minimum.
(d) Because E′(p)=0 guarantees that E′′(p)>0, showing the graph is concave up everywhere and therefore p gives the minimum error.