Step 1: Break the case into the facts that matter:
There are three facts. The headaches look like migraine. A prophylactic drug is wanted. The patient has asthma and mild depression.
The question really asks: which listed drug prevents migraine without harming an asthmatic patient?
Step 2: List the drugs that prevent migraine:
The groups with proven benefit are beta blockers (propranolol, metoprolol, atenolol), tricyclics (amitriptyline), antiepileptics (topiramate, valproate), calcium channel blockers (flunarizine) and CGRP antibody drugs in resistant cases.
Carbamazepine and duloxetine are not on this list.
Step 3: Remove drugs that are not migraine prophylactics:
Carbamazepine is an antiepileptic, but its use is for epilepsy, trigeminal neuralgia and mania. It is not a recommended migraine preventer.
Duloxetine is an SNRI for depression and neuropathic pain. It is not a recommended migraine preventer.
Step 4: Remove the drug that is unsafe for this patient:
Of the two remaining drugs, propranolol and topiramate, propranolol blocks beta-2 receptors. In an asthmatic this can trigger bronchospasm, so asthma is a contraindication.
Topiramate has no action on bronchial tone.
Step 5: Reasoning for the final pick:
Topiramate prevents migraine, is safe in asthma, and is also used in mood disorders such as bipolar illness. So it fits the case best.
Step 6: Practical points about topiramate:
Common side effects are tingling in the hands, weight loss, drowsiness and word finding trouble. It can cause kidney stones and acute angle closure glaucoma. It should be avoided in pregnancy because it can cause cleft lip and palate.
Final Answer:
Topiramate is the migraine preventer that is safe in an asthmatic patient. Propranolol is ruled out by asthma, and carbamazepine and duloxetine are not migraine preventers. The answer is option (C).