A certain country uses a progressive tax system. The amount of tax consists of a linear part proportional to the income and a nonlinear part depending on the income by a power law. The total amount of tax is determined by the formula: T(w)=w+(w+)p. where w is the income, p is an exponent, and , and are positive constants. (a) Calculate at what level of income the tax rate will be minimal. [4 marks] (b) The government wants to double the level of income that is taxed at the minimum rate, but does not want to change the taxes paid by individuals without income. Find a new value for the parameters , and p to achieve the government goals. [3 marks] (c) Explain, in your own words, why the Lagrangian multipliers for inequality constraints must be zero if the constraints are not satisfied strictly at the optimum. [3 marks]