Evaluate:

A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest and has space for at most 20 items. A fan costs him Rs. 360 and a sewing machine Rs. 240. His expectation is that he can sell a fan at a profit of Rs. 22 and a sewing machine at a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize the profit? Formulate this as a linear programming problem and solve it graphically.

If the sum of the hypotenuse and a side of a right-angled triangle is given. Show that the area of the triangle is maximum when the angle between them is

OR

A manufacturer can sell x items at a price of Rs. each. The cost price of x items is Rs. Find the number of items he should sell to earn maximum profit.

Find the value of y, if

If y = 3e^2x + 2e^3x, prove that