= Solution
If $d=1$, $C$ is a line and has no <singular points>. Suppose $d>1$ and take $P$ to be an <irreducible polynomial>. First, $P_x$ is not the zero <polynomial>: otherwise $P\in\mathbb C[y,z]$, and a <homogeneous polynomial> in two variables of degree greater than one factors into linear factors over $\mathbb C$, contrary to irreducibility.
Let $D=V(P_x)$ be the <polar curve> in this coordinate direction. Since $\deg P_x=d-1<d$, the <irreducible polynomial> $P$ cannot divide $P_x$. Thus $C,D$ have no common <irreducible component>, and <Bézout's theorem> applies. One may count repeated factors of $P_x$ with their multiplicities; alternatively, passing to its reduced equation only decreases $\deg D$.
Every <singular point> $p$ of $C$ satisfies $P_x(p)=0$ and has $I_p(C,D)\geq2$ by part (a). Consequently, if $s$ is the number of distinct <singular points>,
$$
2s\leq\sum_{p\in C\cap D}I_p(C,D)\leq d(d-1),
\qquad
\boxed{s\leq\frac{d(d-1)}2.}
$$
This argument also proves that the <singular locus> is finite.
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