Consider SVM classification as illustrated in Figure 7.7. The goal of this exercise is to classify the training points \(\left\{\left(\boldsymbol{x}_{i}, y_{i}\right)\right\}\) based on the value of the multipliers \(\left\{\lambda_{i}\right\}\) in Exercise 10.

Let \(\xi_{i}\) be the auxiliary variable in Exercise \(10, i=1, \ldots, n\).

(a) For \(\lambda_{i} \in(0,1)\) show that \(\left(x_{i}, y_{i}\right)\) lies exactly on the decision border.

(b) For \(\lambda_{i}=1\), show that \(\left(\boldsymbol{x}_{i}, y_{i}\right)\) lies strictly inside the margins.

(c) Show that for \(\lambda_{i}=0\) the point \(\left(x_{i}, y_{i}\right)\) lies outside the margins and is correctly classified.