The demand function for a commodity is \(p={4\over x}\), where p is unit price. Find the instantaneous rate of change of demand with respect to price at p = 4. Also interpret your result.

The cost function of a firm is \(C={1\over3}x^3-3x^2+9x\). Find the level of output (x > 0) when average cost is minimum.

Find the price elasticity of demand for the demand function x = 10 – p where x is the demand and p i the price. Examine whether the demand is elastic, inelastic or unit elastic at p = 6.

Find the equilibrium price and equilibrium quantity for the following functions. Demand: x = 100 – 2p and supply: x = 3p – 50

If f (x, y)  = 3x² + 4y³ + 6xy - x³y³ + 7 then show that f_xy (1, 1) = 18.

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

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

Find the maximum and minimum values of x³-6x²+7

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

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