mathematical economics calculus matrix algebra and optimization problems

Please See Attachment for clarity – All work/steps to solution must be shown

Simplify the following expression so that it contains only a single exponent of x [8 points]

[8 points] Find Y* (equilibrium output) from the following [Hint: Y=(Y^1/2)²]:

[8 points] Find the maximum profit and the quantity (Q) that will maximize the profit function below.Show evidence that profit is indeed maximized:

[8 points] Find the price elasticity of demand of the following:
[8 points total] Find the derivatives of the following by first taking the natural logs:
[ points] Given the following matrices:
[8 points]Assume a stock price is a positive function of earnings per share [E], a negative function of bond interest rate [i], and a positive function of inflation [Ï€] and GDP [Y].The function is denoted:S = S(E, i, Ï€, Y).To complicate matters GDP is a negative function of bond interest rate.Write the total differential equation.What can you say about the total derivative of S with respect to the bond interest rate?

[8 points] Find the critical values of the following and use the N^th derivative test to determine if the values determine a maximum, minimum or inflection point.

[8 points] Find the values of x, y and that maximize the following utility function subject to the budget constraint: ;

Find, or if it does not exist state that it does not exist:

Find x₁ using Cramerâ€™s rule.