C$.$ Theodore Twombly is producing a ratherly unusual service. He runs a business that has

professional writers and artificial intelligence compose love letters for people who are unable

to write them to their lovers themselves. It is a seemingly labor$-$intensive business, however,

Theodore put more emphasis and maintenance costs on Al than human writers so that the

cost function of his business mathematically takes the form:

$C=15A+5H$

while his business is constrained to a production function as follows:

$A^{0\mathrm{.}6}{H}^{0\mathrm{.}4}=132$

A and $H$ represents $Al$ and human input of production, respectively. Help Theodore Twombly

to optimize his business by minimizing his production costs:

ii$)$ What is the optimal value of A and H$?$

iii$)$ Show that your answers in $($ii$)$ are indeed minimizing the product cost.