How could you identify whether a particular bacterial sample contained specimens with mycolic acid-rich cell walls?

By the application of an acid-fast staining technique.

Explanation:

Mycolic Acid Cell Wall:

Mycolic acid is a long chain fatty acid abundantly found in the cell wall of of bacterium belonging to the genus Mycobacterium e. g. Mycobacterium tuberculosis. Mycolic acids comprise about 40 to 60% of the mycobacterial cell wall. These bacteria are called acid fast bacteria because their cell walls cannot be stained with Gram staining technique.

Mycolic acid makes the mycobacterial cell wall so thick that regular Gram stains cannot penetrate it. Therefore, a more advanced acid fast staining technique is used.

Acid Fast Staining:

Acid fast staining is a differential staining technique comprising of the Ziehl-Neelsen technique and the Kinyoun technique. Both involve the use of carbolfuchsin stain, a lipid soluble, phenolic dye that penetrates through the mycobacterial cell wall. The cell wall retains the dye even after decolorization.

After the application of carbolfuchsin, a counter stain methylene blue is used that stains the non-acid fast bacteria without mycolic acid cell walls.