Question

6. Find the maximum and minimum values of the following functions

a) 3X4 -X3 +2

b) x4 – 14x2 +24x +9

7. Find the profit maximizing output given Q = 200 – 10p and AC = 10 + Q25 where Q is quantity, p is price and AC is average cost.

8. Find the first order and second order partial derivatives of the following function

a) Z = 2x3 +5 x2y +xy2 +y3

b) Z = log (x2 + y2)

9. Find elasticity of demand if demand function is x = 250 – 5p +p2. Also find elasticity of demand at p = 8

10. Find elasticity of demand if demand function is p = 50 – 3q. Also find elasticity of demand at p = 5

Answer 1

a)

$\frac{\partial z}{\partial x}=6x^2+10xy+y^2,$

$\frac{\partial z}{\partial y}=5x^2+2xy+3y^2,$

$\frac{\partial ^2z}{\partial x^2}=12x+10y,$

$\frac{\partial^2 z}{\partial y^2}=2x+6y^2,$

$\frac{\partial ^2}{\partial x\partial y}=10x+2y,$

b)

$\frac{\partial z}{\partial x}=\frac{2x}{x^2+y^2},$

$\frac{\partial z}{\partial y}=\frac{2y}{x^2+y^2},$

$\frac{\partial^2 z}{\partial x^2}=\frac{2(y^2-x^2)}{(x^2+y^2)^2},$

$\frac{\partial^2 z}{\partial y^2}=\frac{2(x^2-y^2)}{(x^2+y^2)^2},$

$\frac{\partial^2 z}{\partial x\partial y}=\frac{4xy}{(x^2+y^2)^2}.$