(Solved): all of the details are here Fred's flower shop creates and sells two types of speciality bouquets. ...

Fred's flower shop creates and sells two types of speciality bouquets. One is a native bush-flower bouquet and the other is a tropical delights bouquet. Let \( x \) be the number of native bush-flower bouquets (in tens of units) and \( y \) be the number of tropical delight bouquets (in tens of units) sold per month. Fred's profit is a function of these two variables. Fred knows that the monthly demand for his native bushflower bouquet is given by \( 80-4 x \) dollars whilst the monthly demand for the tropical delights bouquet is given by \( 60-2 y \) dollars. He also knows that the joint cost of producing these two bouquets is \( 4 x y-4 \) (in tens of dollars) each month. (a) Determine Fred's monthly profit function \( P(x, y) \). (b) Find any critical point(s) that satisfy both \( P_{x}=0 \) and \( P_{y}=0 \). (c) Use the second derivative test to determine how many of each bouquet Fred must sell to maximise his monthly profit. (d) Calculate Fred's maximum monthly profit. Fred's flower shop creates and sells two types of speciality bouquets. One is a native bush-flower bouquet and the other is a tropical delights bouquet. Let \( x \) be the number of native bush-flower bouquets (in tens of units) and \( y \) be the number of tropical delight bouquets (in tens of units) sold per month. Fred's profit is a function of these two variables. Fred knows that the monthly demand for his native bushflower bouquet is given by \( 80-4 x \) dollars whilst the monthly demand for the tropical delights bouquet is given by \( 60-2 y \) dollars. He also knows that the joint cost of producing these two bouquets is \( 4 x y-4 \) (in tens of dollars) each month. (a) Determine Fred's monthly profit function \( P(x, y) \). (b) Find any critical point(s) that satisfy both \( P_{x}=0 \) and \( P_{y}=0 \). (c) Use the second derivative test to determine how many of each bouquet Fred must sell to maximise his monthly profit. (d) Calculate Fred's maximum monthly profit.