How can I show that both x and y are both bounded, for the below question ?

Please explain clearly showing each step as thoroughly as possible. If you are using hand written notes, then please ensure they are neat and legible, as it is difficult to interpret illegible hand-written notes. Alternatively, use LaTex.

Question

A rm can sell two goods, X and Y, to receive a total prot of H(:r,y) =aln(3:+ 1) + (1 a)ln(y +1)1 where :1: and y are the quantities sold of goods X and Y respectively, and a is a positive constant. To produce goods X and Y, the rm requires just two inputs labour and capital such that 0 one unit of good X requires 3 units of labour and 4 units of capital, and 0 one unit of good Y requires 4 units of labour and 3 units of capital. If we suppose that the rm has both 17 units of labour and 17 units of capital, then the production constraints the rm faces are 3$+4y 17, 4m+3y$ 17. The rm's prot maximisation problem is therefore to maximise H on the feasible set D= {(x,y)eR2| 2:20, we 3x+4ys 17, 4x+3ys 17}. (8.) Explain why this inequality constraint optimisation problem has a solution. State any results that you use