$[25$ points$]$ Consider a consumer with M $=$ $$9$ to spend $($income$)$ and faces prices PX $=$ $$6$ and PY $=$ $$3$ forgoods X and Y$.$ Her utility function is U$($X$,$ Y$)=$ X$1/2+$ Y$.$a $[5$ points$]$ Carefully express the consumers choice problem, using the given information $($this is whereyou write out the max operator, the choice variables, the objective function, and the budget constraint$).$b $[10$ points$]$ Compute the absolute value of the consumers marginal rate of substitution, and inspect it todetermine the shape of the consumers indifference curves: C$-$shaped, linear, $)-$shaped, or some other shape.To show your work, neatly use the arrow argument, increasing X $()$ and decreasing Y $()$ to see whether$|$MRS$($X$,$ Y$)|$ is diminishing along an indifference curve.c $[10$ points$]$ If the indifference curves are C$-$shaped write out the budget line and the equal slopes conditionthat characterize an interior solution to the consumers choice problem. Use the particulars for the givenconsumer. Solve these conditions to find the interior solution. On the other hand, if the indifference curvesare linear or $)-$shaped, look for a boundary solution where the consumer either buys all X or all Y$.$ Determinewhich boundary is the solution by comparing the consumers utility at $($M$/$PX$,0)$ and $(0,$ M$/$PY$),$ which arethe corners of the budget line.