Firms are motivated to minimize production costs because:
OCW Scholar - Principles of Microeconomics
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...because...the assumption for Producer Theory is that firms have the main goal to maximise profits and not market position. Since profits are the revenue minus the total costs, then it would be logical to minimise costs to maximise profits.
You need to remember that any firm is inteterested in getting more revenue. As you possibly know profit=price*quantity - cost f(quantity). if you look at this simple equation you can easily figure out the answer of your question. if the firm minimize the cost, then this equation says it will increase its profit. Equation says as well if the firm sell at higher price the profit will increase (as far the quantity is the same).
In short minimizing the cost or maximizing the profit is making the firm better.
Important note, profit maximizing firm is a cost minimizing, but cost minimizing firm not necessarly to be profit maximizing