provide ,solutions for these questions.

1. Solve the problem, min f(x,y)=24x-x3 +16y-2y subject to g(x, y)= x +2y =44 a. Verify that (x*, )*) solves this minimization problem. b. What is the approximate change in the optimal value of /(x, y) if 44 is changed to 45?(i) Describe briefly the Vasicek one-factor model of interest rates and its key statistical properties. [4] (ii) According to a particular parameterisation of this model, the instantaneous forward rate applicable at a fixed time T >t implied by the market prices at time t is found to be: f(t,T)=near +(1-e-")+k(1-e-")e-ar where r = T-t and a > 0. Show that, if a humped curve is required for f(t,T), the parameter values must satisfy the condition k> |0 - 4. [5] Here a "humped curve" means one where the value of the function for some intermediate values of T exceeds the values for both : =0 and r = co. In other words, there will be a maximum value for some positive value of r . (iii) Describe briefly the main advantages and limitations of the Vasicek model. [4] [Total 13] Claims occur according to a compound Poisson process at a rate of % claims per year. Individual claim amounts, X , have probability function: P(X =50) =0.8 P(X =100) =0.2 The insurer charges a premium at the beginning of each year using a 20% loading factor. The insurer's surplus at time t is U(t). Find P[U(2) 0? when f(y)