Making sense of the notation of Chain of critical points

Question

This could be a silly question but nevertheless i am confused with the book i am referring to , the statement goes like this First Notations : $$C_i=\{x\in U : \frac{\partial^{j_1}....\partial^{j_n}}{\partial x_1^{j1}....\partial{x_n}^{j_n}}f(x)=0\} $$ for all $j$ with $j_1+j_2+....+j_n \le i$

The claim is $C_i \subset ....... C_1 \subset C_o$

But this doesn't seem to hold true by taking simple example $f(x)=3x+x^2$ , $f'=2x+3$

$-3/2 \in C_1$ but not in $C_0$ I think my understanding is false here .

Answer 1

Think of a single variable. Then $$ C_0 = \{x \in U \mid f(x)=0\}, $$ while $$ C_1 = \{x \in U \mid f(x) =0 \text{ and } f'(x)=0\}. $$