The elasticity of demand for the demand function x = \(\frac { 1 }{ p } \) is______.

Relationship among MR, AR and η_d is ______.

If u = 4x² + 4xy + y² + 32 + 16 , then \(\frac { \partial ^{ 2 }u }{ \partial y\partial x } \) is equal to ________.

If u = x³ + 3xy² + y³ then  \(\frac { \partial ^{ 2 }u }{ \partial y\partial x } \) is _______.

if q = 1000 + 8p₁ - p₂ then, \(\frac { \partial q }{ \partial { p }_{ 1 } } \) is _______.

If y=x-1/x, prove that y is a strictly increasing function for all real vaules of x(x\(\neq\)0).

If y=1+1/x, show that y is a strictly decreasing function for all real values of x(x\(\neq\)0).

Prove that 75-12x+6x²-x³ always decreases as x increases.

If f(x,y) = 3x² + 4y³ + 6xy - x²y³ + 6. Find f_yy(1,1)

If f(x,y) = 3x² + 4y³ + 6xy - x²y³ + 6. Find f_xy(2,1)

Find the stationary points and stationary values of the function f(x) = x³ - 3x² - 9x + 5.

Show that the function x³ + 3x² + 3x + 7 is an increasing function for all real values of x.

Separate the intervals in which the function x³ + 8x² + 5x - 2 is increasing or decreasing.

Find the maximum and minimum values of the function x² + 16/x

For the production function P= 5(L)^0.7(K)^0.3.Find the marginal productivities of Labour (L) and Capital (K) when L = 10, K = 3 [Use (0.3)^0·3 = 0.6968; (3.33)^0·7 = 2.2322]

A firm has revenue function R = 8x and production cost function \(C = 150000 + 60\left(x^2\over 900\right)\) Find the total profit function and the number of units to be sold to get the maximum profit.

Verify Euler's theorem for the function \(u=\sqrt{x^2+y^2}\)

If \(u= e^{x/y} sin\left(x\over y\right)+e^{y/x}cos\left(y\over x\right)\) show that \(x{∂u\over ∂x}+y{∂u\over ∂y}=0\) using Euler's theorem.

For the production function P = C(L)^α(K)^β where C is a positive constant and if α + β = 1, show that \(K{∂P\over ∂ K}+L{∂P\over ∂L}=P\)