QUESTION TWO $(33$ MARKS$)$

a$)$ Find the relative extremum of the functions $f(x,y)=e^{2x}-2x+2y^2+3$ and identify whether the extremum is relative maximum or minimum using second order conditions.

$(5$ marks$)$

b$)$ To reduce shipping distances between the manufacturing facilities and a major consumer, a Korean computer brand, Intel Corp. intends to start production of a new controlling chip for Pentium III microprocessors at their two Asian plants. The cost of producing $x_1$ chips at Chiangmai $($Thailand$)$ is $C_1=0\mathrm{.}002x_1^2+4x_1+500,$ and the cost of producing $x_2$ chips at Kuala$-$Lumpur $($Malaysia$)$ is $C_2=0\mathrm{.}005x_2^2+4x_2+275.$ The Korean computer manufacturer buys them for RM$150$ per chip.

i$)$ Find the quantity that should be produced at each Asian location to maximize the profit.

$(8$ marks$)$

ii$)$ Using the second order condition $($SOC$),$ prove that the profit is at the maximum.

$(3$ marks$)$

c$)$ A firm produces two goods in quantities $x$ and $y.$ Its cost function is $C(x,y)=10x+xy+10y$ and the prices $P_x$ and $P_y$ it can charge are, respectively, $P_x=50-x+y$ and $P_y=50-x+y.$ The firm is committed to delivering a total of $15$ units. How much should the firm produce of each good to maximize profits?

$(10$ marks$)$

d$)$ Maximise $f=x-y^2$ subject to $x^2+y^{2}4,$ where $x,y0$

i$)$ Write the Lagrangian function.

$(2$ marks$)$

ii$)$ Write the Kuhn$-$Tucker conditions for this maximization problem.

$(5$ marks$)$