Answer to Question #214926 in Calculus for wweeee

Question

If the productivity of a company is represented by the function

f(x,y)=40x^3/4y^1/4 with the utilization of x units of labour and y units of capital. Find

i) F_x and F_y and interpret

ii)If today the company is using 400 units of labour 250 units of capital. Determine the change in productivity if the labour is increased by 12 units and the capital is decreased by 8units.

Answer 1

i)

$$f_x=\dfrac{\partial f}{\partial x}=30x^{-1/4}y^{1/4}$$

This is the marginal productivity of labor.

$$f_y=\dfrac{\partial f}{\partial y}=10x^{3/4}y^{-3/4}$$

This is the marginal productivity of capital.

ii)

$$df=30x^{-1/4}y^{1/4}dx+10x^{3/4}y^{-3/4}dy$$

$$\Delta f=30(400)^{-1/4}(250)^{1/4}(12)+10(400)^{3/4}(250)^{-3/4}(-8)$$

$$\approx206$$