Consider a firm with long$-$run production function

Q$=$f$($L$,$K$)=3$L$^(1/2)$K$^(1/2),$

where Q is output, L measures labor input, and K measures capital input.

Input prices are w $=12$ and r $=3$ per unit of labor and capital, respectively.

At the given factor prices, what is the firm$$s $($long$-$run$)$ cost as a function of output, C$($Q$)?$

Compute the firm$$s $($long$-$run$)$ marginal cost and average cost curves at the given factor prices, MC$($Q$)$ and AC$($Q$),$ respectively?

In the short$-$run capital is fixed at level K$=16.$ Find the firm$$s short$-$run total cost, short$-$run marginal cost, and short$-$run average cost curves, STC$($Q;K $=16),$ SMC$($Q;K $=16),$ SATC$($Q;K $=16),$ respectively?

Now consider some comparative statics properties as parameters change $($only change one parameter at a time and consider the other parameters unchanged from the problem above$).$

How would your answers to parts $1$ and $2$ change if the firm faced input price r$\text{'}=9$ for capital instead of r $=3?($Here w $=12$ remains unchanged$)$

How would your answers to parts $1$ and $2$ change if the firm faced input price w$=3$ for labor instead of w $=12?($Here r $=3$ remains unchanged$)$

How would your answers to part $3$ change if the firm$$s capital level were fixed at K $=9$ instead of K $=16?($Here w $=12$ and r $=3$ remain unchanged$)$