A concert promoter produces two kinds of souvenir shirt, one kind sells for $18 ad the other for$25. The company determines, the total cost, in thous

Multivariable functions
A concert promoter produces two kinds of souvenir shirt, one kind sells for $18 ad the other for$25. The company determines, the total cost, in thousands of dollars, of producting x thousand of the $18 shirt and y thousand of the$25 shirt is given by
$$\displaystyle{C}{\left({x},{y}\right)}={4}{x}^{{2}}-{6}{x}{y}+{3}{y}^{{2}}+{20}{x}+{19}{y}-{12}.$$
How many of each type of shirt must be produced and sold in order to maximize profit?

2021-01-06
Revenue function
$$\displaystyle{R}{\left({x},{y}\right)}={18}{x}+{25}{y}$$
Profit function
$$\displaystyle{C}{\left({x},{y}\right)}={R}{\left({x},{y}\right)}-{C}{\left({x},{y}\right)}$$
$$\displaystyle{C}{\left({x},{y}\right)}=\cdot{18}{x}+{25}{y}{)}-{\left({4}{x}^{{2}}-{6}{x}{y}+{3}{y}^{{2}}+{20}{x}+{19}{y}-{12}\right)}$$
$$\displaystyle{C}{\left({x},{y}\right)}=-{4}{x}^{{2}}+{6}\times{y}-{3}{y}^{{2}}-{2}{x}+{6}{y}+{12}$$
First we will find the critical point. Now,
$$\displaystyle{P}_{{x}}{\left({x},{y}\right)}=-{8}{x}+{6}{y}-{2}$$
$$\displaystyle{P}_{{y}}{\left({x},{y}\right)}={6}{x}-{6}{y}+{6}$$
$$\displaystyle{P}_{{\times}}{\left({x},{y}\right)}=-{8}$$
$$\displaystyle{P}_{{{y}{y}}}{\left({x},{y}\right)}=-{6}$$
$$\displaystyle{P}_{{y}}{\left({x},{y}\right)}={0}$$
6x-6y+6=0
y=x+1
And $$\displaystyle{P}_{{x}}{\left({x},{y}\right)}={0}$$
6y-8x-2=0
6(x+1)-8x=2
x=2 (i)
So, y=x+1
y=2 (ii)
Hence, the critical point is (2,3)
Since $$\displaystyle{f}_{{\times}},{f}_{{{y}{y}}},{\quad\text{and}\quad}{f}_{{{x}{y}}}$$ all are constant so we will need to pkug critical point to function
$$\displaystyle{D}{\left({x},{y}\right)}={f}_{{\times}}{f}_{{{y}{y}}}-{{f}_{{{x}{y}}}^{{2}}}$$
$$\displaystyle={\left(-{8}\right)}{\left(-{6}\right)}-{\left(-{6}\right)}^{{2}}$$
=12>0
Since, D>0, and $$\displaystyle{f}_{{\times}}{<}{0}$$</span>
The critical point P(2,3) has a relative maximum.
And the maximum value of profit function
$$\displaystyle{P}{\left({2},{3}\right)}=-{4}{\left({2}\right)}^{{2}}+{6}\cdot{2}\cdot{3}-{3}{\left({3}\right)}^{{2}}-{2}\cdot{2}+{6}\cdot{3}+{12}$$
$$\displaystyle{P}{\left({2},{3}\right)}={19}$$
The maximum profit of $19000 will be earned if 2000 shirt of$18 and 3000 shirts of \$25 are produced and sold.