 Solution

The entropy chain rule givesConditional entropy is nonnegative, so $I(X;Y)\le \min (H(X),H(Y))$. For finite entropies, if $H(X)\le H(Y)$, equality holds exactly when $H(X\mid Y)=0$, meaning $X$ is a function of $Y$ almost surely. If $H(Y)\le H(X)$, interchange the variables. When their entropies are equal, equality requires both functional relations. Zero conditional entropy has this characterization because each conditional probability distribution must be a point mass.