a , b ] if x f x f x & $ for all x x a b e @ z A function f ( x ) is said to be strictly decreasing function in [ a , b ]if f x f x > & h for all x x a b e @ z Let f be a differentiable function on an open interval ( a,b ) containing c and suppose that '’( ) c exists. (i) If f '( c ) = and f "( c ) > , then f has a local minimum at c . (ii) If f '( c ) = and f "( c ) < ,then f has a local maximum at c . z A function f ( x , y ) of two independent variables x and y is said to be homogeneous in x and y of degree n if for t > f tx ty t f x y ( , ( , ) z If u f x y ( , ) is a homogeneous function of degree n , then x u y u nu ∂ ∂ + ∂ ∂ z The partial elasticity of demand q with respect to p is defined to be η qp Eq Ep q q = − ∂ ∂ z The partial elasticity of demand q with respect to p is defined to be η qp Eq Ep q q = − ∂ ∂ Some standard results .
Total cost: C x f x k ( ) ( ) . Average cost: AC f x k c x ( ) ( ) . Average variable cost: AVC= f x ( ) . Average fixed cost: AFC k .
Marginal cost: MC dC . Marginal Average cost: MAC dx AC . Total cost: C x AC ( ) = × . Revenue: R px .
Average Revenue: AR - - . Marginal Revenue: MR dR . Profit: P x R x C x ( ) ( ) (