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Q6. A tech start-up company has a monthly advertising budget of $60,000. Its marketing department estimates that if :1: dollars are spent each month on advertising in newspapers, y dollars per month on television advertising, and .3 dollars per month on online advertising. then the monthly sales will be given by S = 40miyfzi: dollars. The profit is 10% of the sales. (3) Write down the optimisation question that the company faces to maximise its profits. [5 points] (b) Write down the Lagrangian equation and the first-order con- ditions. then find the optimum amounts of dollars that needs to be spent on newspaper, television, and online advertising to maximise the profits. [10 points] (c) Show that at the optimum allocation, returns to each dollar spent on each advertising form brings the same returns to the company. [6 points] (d) What's the interpretation of the Lagrange multiplier in this case? Please be concise and express in your own words. [6 points] (e) If the advertising budget of the company expands by 25%, how does this affect the optimal allocation of its budget on different advertising forms and its profits. Please provide a discussion. [8 points]