QUESTION TWO $(20$ MARKS$)$

a$)$ Given $Z=2x^2+y^2-12x-8y+50$

i$)$ Find the critical values.

$(4$ marks$)$

ii$)$ Prove that the critical values are at a relative maximum$/$ minimum.

$(4$ marks$)$

b$)$ An individual's utility function is given by

$U=260x_1+310x_2+5x_1{x}_2-10x_1^2-x_2^2$

where $x_1$ is the amount of leisure measured in hours per week and $x_2$ is earned income measured in RM per week.

i$)$ Find the utility$-$maximising values of $x_1$ and $x_2.$

$(6$ marks$)$

ii$)$ Calculate the maximum utility.

$(2$ marks$)$

iii$)$ Use the Hessian, prove that the utility is at a maximum.

$(4$ marks$)$