Ascertain the products $B_1$ and $C_1$ of the following reaction: 




Step 1: Initial reaction with acetic acid.
In the first step, the methyl group (\( \text{CH}_3 \)) undergoes oxidation with acetic acid, forming an intermediate compound where the amino group (\( \text{NH}_2 \)) remains attached to the benzene ring. This is a Friedel-Crafts type alkylation.
Step 2: Oxidation with KMnO\(_4\).
The second step involves oxidation using alkaline KMnO\(_4\). Potassium permanganate is a strong oxidizer that will oxidize the methyl group (\( \text{CH}_3 \)) into a carboxyl group (\( \text{COOH} \)).
Step 3: Final acidification.
The final step involves acidification with dilute \( \text{H}_2\text{SO}_4 \), which helps to stabilize the carboxyl group formed during oxidation.
Conclusion:
The product \(B_1\) is a carboxylated intermediate (\( \text{COOH} \)), and the final product \(C_1\) is the carboxylated compound with a carboxylic acid group.
Let the function $ f(x) $ be defined as follows: $$ f(x) = \begin{cases} (1 + | \sin x |)^{\frac{a}{|\sin x|}}, & -\frac{\pi}{6}<x<0 \\b, & x = 0 \\ \frac{\tan 2x}{\tan 3x}, & 0<x<\frac{\pi}{6} \end{cases} $$ Then the values of $ a $ and $ b $ are: