Need help understanding the following, specifically $($b$)$ and $($c$).$

Consider the following cost function: $c(w_1,w_2,y)=w_1^{\frac{1}{4}}{w}_2^{\frac{3}{4}}{y}^2.$

$($a$)$ Find the marginal cost function, the average cost function, the supply function, and the

demand function for input $1.$

$($b$)$ Sketch the supply curve and the demand curve for input $1.$

$($c$)$ What is the effect of an increase in the price of input $2$ on the supply curve and the

demand curve for input $1?$ Illustrate your answers.

Solution: $($excluding graphs$)$

$MC=\frac{\text{d}\text{e}\text{l}\text{c}}{\text{d}\text{e}\text{l}\text{y}}=2w_1^{\frac{1}{4}}{w}_2^{\frac{3}{4}}y$

$AC=\frac{c}{y}=\frac{w_1^{\frac{1}{4}}{w}_2^{\frac{3}{4}}{y}^2}{y}=w_1^{\frac{1}{4}}{w}_2^{\frac{3}{4}}y$

Supply: $p=MC=>p=2w_1^{\frac{1}{4}}{w}_2^{\frac{3}{4}}y$

To get demand for input $1,$ use Shephard's lemma:

$x_1=c_{w_1}(w,y)=\frac{1}{4}{w}_1^{-\frac{3}{4}}{w}_2^{\frac{3}{4}}{y}^2$