认认真真 is the AABB reduplication of 认真 ('serious / conscientious'), a productive pattern in Chinese for adjectives that intensifies the quality and adds a vivid, emphatic feel. Other examples: 高高兴兴, 干干净净, 漂漂亮亮, 安安静静. Almost always paired with 地 before a verb (认认真真地做).
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Slightly more spoken/colloquial than plain 认真. The reduplication adds warmth and emphasis - common in praise to children: 这孩子做事认认真真的 ('this kid really does things properly').
