$5.(20$ points$)$ Let's assume that the production technology of a firm using labor and capital to produce good x is homothetic. According to this production technology, there are increasing returns to scale when the level of production is between $0$ and xA$,$ constant returns to scale when between xa and xB$,$ and decreasing returns to scale when above xB $(0<$ xA $<$ xB$).$

$($a$)$ Draw isoquants for xA and xB on a graph with labor on the horizontal axis and capital on the vertical axis. For the given input prices w and r$,$ indicate the point A $=($lA$,$ kA$)$ at which you can produce xa at the lowest cost using an isoquant. Indicate the slopes of the isoquant and the isoquant at point A$.$

$($b$)$ On the same graph, show where point B should be located where you can produce xB at the lowest cost. What will be the shape of the vertical section passing through all cost minimization points? Show it on a separate graph with inputs $($l and k together, you can assume l$=$k only for this graph$)$ on the horizontal axis and x on the vertical axis.

$($c$)$ Redraw the graph you drew in $($a$).$ On this graph, mark the possible data of the input baskets for which the firm will maximize its profit for any product price p $>0$ and explain this situation.

$($d$)$ Let $($l$*,$ k$*,$ x$*)$ be the production plan for which the firm maximizes its profit. Which two conditions regarding the marginal efficiency of inputs will be satisfied in this production plan $($and only in this production plan$)?$ Explain.