Let be the interest paid by the bank.

The amount of money deposited by bank at percent interest and minus of the interest paid to the customer.

The amount of the money deposite is proposinal to the square of the interest rate.

Assume .

Substitute in .

Differentiate on each side with respect to .

Evaluate to find maximum interest.

Interest rate of bank that should be paid for maximum profit is .

Interest rate of bank that should be paid for maximum profit is .