Does the marginal rate of substitution (MRS) diminish along the indifference curve

Mathematics by Geoffrey
  1. An indifference curve is given by X0.4Y0.7 = 100, where X and Y are quantities of goods X and Y consumed, with Y is on the vertical axis and X is on the horizontal axis.
  2. Find the slope of the indifference curve.
  3. Does the marginal rate of substitution (MRS) diminish along the indifference curve

when X increases?  Prove your answer.

A generalized production function Q=f(K,L) is defined implicitly from the following equation, f(Q, K, L) = QerQ-AKaLb = 0. Find the marginal product of L and K, and the MRTS

3. A demand function for good X is given by X=APxaPyβIγ,

  1. Given A=  and      B=.
  2. Find AB and BA, showing AB ¹
  3. Show that det(AB) = det(BA) = (detA)*(detB).
  4. Find A+B and show that det(A+B) ¹ det(A)+det(B).