The binary cross$-$entropy loss function is defined by

$l(p,q)=-$plog$(q)-(1-p)log(1-q)$

for any numbers $p$ and $q$ with $0p1$ and $log(q)qp=\mathrm{.}75l(p,q)qpql(p,q)np.log0.$ Here $log(q)is$ the

natural logarithm $ofq.$

$(a)$ What $is$ the binary cross$-$entropy function used for $in$ machine learn$-$

ing?

$(b)$ Let $p=\mathrm{.}75$ and use matplotlib.pyplot $to$ plot $l(p,q)as$ a function

$ofq,$ with $p$ held fixed. What value $ofq$ minimizes $l(p,q)?$

Hint: The function $np.logis$ useful.