Final answer:
For an infant with signs of shock and hypoglycemia, the treatment should involve rapid IV fluid resuscitation and correction of hypoglycemia with IV dextrose, along with the investigation and treatment of the underlying cause.
Step-by-step explanation:
The most appropriate treatment for an infant presenting with vomiting, diarrhea, weak central pulses, and hypoglycemia (glucose at 30mg/dL) includes aggressive supportive care. Given the clinical signs of shock (cool extremities, capillary refill time of >5 sec, rapid and weak pulses), initial management should involve rapid intravenous fluid resuscitation using isotonic crystalloid fluids, such as 0.9% saline or Ringer's lactate. After securing intravenous access, which has already been done as per the scenario, the hypoglycemia needs to be corrected urgently with IV dextrose to prevent further deterioration of the infant's condition and potential brain injury. In addition to these interventions, the underlying cause of the infant's symptoms, which could include infection or another illness provoking the vomiting and diarrhea, should be investigated and treated accordingly.