Please show full math and an answer for each part!

Q3.a (6 points) Rodrigo makes vegan ice cream. His production function for the number of ice creams he makes is f(x1 , x2) = 8(x] + x; ) where x, is the amount of almond milk, and x2 is the number of dry fruits. His standard rate for a large scoop of vegan ice cream is $6. Almond milk costs $1, and dry fruits cost $3 per scoop. There is an additional fixed cost of $2 for rent that he has to pay regardless of whether any ice cream is produced or not. (a) Given the prices of inputs, what is the profit-maximizing input? How many ice cream scoops should Rodirgo sell to maximize his profit? What would be his profit? (b) The government introduces a corporate tax of 20%. What is Rodrigo's after-tax profit? How much of each input will he be using now? (c) Instead of introducing the 20% corporate tax in part (b), the government decided to add an ad valorem input tax of 100% to input 2. What is the profit-maximizing amount of each input to use? What is the profit-maximizing output? How much is Rodrigo's new profit