Question in computer science sir solve this problem and provide me correct solution I$\text{'}$m in very hurry...

Question $1$ Consider the following production function:

$y=F(L,K)=4L^{\frac{1}{t}}{K}^{\frac{1}{3}}$

where $L$ and $K$ are the amount of labour and capital used in the production process, and $y$ is the output. Throughout this question, the output price $p$ is $3$ and the rental rate of capital $r$ is $1.$

We will first consider a firm in the short run, where the amount of capital is fixed at $K=64.$ The fixed cost is therefore $64.$

$($a$)($Level A$)$ Is there diminishing returns to labour? Explain.

$($b$)($Level A$)$ Suppose the wage rate $w$ is $1.$ Find the profit$-$maximising choice of $L.$ Calculate the profit$-$maximising output level and the maximised profit. $($There is no need to check the second order condition $-$ but of course you can check if you want to$.)$

$($c$)($Level A$)$ Now suppose $w$ increases to $2.$ Find the profit$-$maximising choice of $L.$ Calculate the profit$-$maximising output level and the maximised profit. $($There is no need to check the second order condition.$)$ You can leave your answers in square roots.

$($d$)($Level A$)$ What is the change in $L$ when $w$ increases from $1$ to $2$ in the short run? You can leave your answers in square roots.

Question $2[36$ marks$]$

A mass of consumers is uniformly distributed along the interval $0,1.$ Two firms, A and B$,$ are located at points $0$ and $1$ respectively. We denote by $p,$ the price of firm iinA,$B.$ A consumer located at point xin$[0,1]$ obtains utility $U_A(x)=u-p_A-t{x}^2$ if he consumes from firm $A,$ and $U_B(x)=u-p_H-t(1-x{)}^2$ if he consumes from firm B$.$ In the following, we assume that the groes utility $u$ is sufficiently high, so that the market will be covered and all consumets will get positive utility in equilibrium. Both firms have a cost function equal to $T_i(q_1)=(1+x)q_i,$ where you should substitute $x$ for the last number of your student ID number.

$($a$)$ Find the demand function for both firms.

$[5$ marks$]$

$($b$)$ Assume firms set their prices simultaneously. Solve for the Nash equilibriuma prices, and compute the equilibrium profits. $[6$ marks