Question

^{(a) Find fx(x,y), fy(x,y), fx(1,3), and fy(-2,4) for the given function. If}

^{𝑧 = 𝑓(𝑥, 𝑦) = 3𝑥ଷ 𝑦ଶ − 𝑥ଶ 𝑦ଷ + 4𝑥 + 9}

^{(b) A firm estimates that it can sell Q units of its product with an advertising expenditure of x thousand dollars where}

^{𝑄 = 𝑄(𝑥) = −𝑥ଶ + 600𝑥 + 25}

^{i) Over what level of advertising expenditure is the number of units of product sold increasing?}

^{ii) Over what level of advertising expenditure is the number of units of product sold decreasing?} 

Answer 1

a)

$z=3x^3y^2-x^2y^3+4x+9$

$f_x=9x^2y^2-2xy^3+4$

$f_x(1,3)=9\cdot9-2\cdot27+4=31$

$f_y=6x^3y-3x^2y^2$

$f_y(-2,4)=-6\cdot8\cdot4-3\cdot4\cdot16=-384$

b)

$Q(x)=-x^2+600x+25$

$Q'(x)=-2x+600=0$

$x=300$

i)

$Q'(x)>0$ for $0\le x<300$ : the number of units of product sold increasing

ii)

$Q'(x)<0$ for $x>300$ : the number of units of product sold decreasing