Suppose you are training a softmax regression model with three classes, and the true label for a given instance $y=C.$ The model's predicted probabilities for this instance are

$P(y=A)=0\mathrm{.}1,P(y=B)=0\mathrm{.}2,P(y=C)=0\mathrm{.}7.$

$($a$)$ Write the cross$-$entropy cost function for this multi$-$class problem.

$($b$)$ Compute the cross$-$entropy cost for this instance.

$($c$)$ Which class the highest predicted probability, and what is its value?