If the production function is F(K,L)=A∗K0.4L0.6 then what is the marginal rate of technical substitution

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Juan Valdez owns a coffee farm in Colombia. His production function is:  f(x1​,x2​)=(x1​−1)^0.25 x2^0.5​   Assume the price of input 1 is r and the price of input 2 is w. (a)  Write down an expression for the technical rate of substitution. (b)  Find Juan's demand for inputs conditional on the quantity y of coffee Juan wants to produce. (c)  Find Juan's cost function. (d)  What is the supply function of Juan's firm?

Cheburashka uses kiwi fruits (K) and labour (L) to produce juice (q). His production function is: q=min{L/3, K0.5}, where labour is measured in hours, kiwi fruits in kg, and juice in (large) bottles. For example, if he uses 4 kg of kiwi fruits and 6 hours of labour, he can produce 2 bottles of juice. a) Draw isoquants for q = 1, q = 2 and q = 3 on a graph with labour on the horizontal axis and kiwi fruits on the vertical one. = $5 per kg. b) Let the price of labour-hour be given by w = $18 and the price of fruit by p What is the optimal amount of labour and fruit to produce q bottles of juice if Cheburashka strives to minimise costs? (Note that you need to provide expressions L(q) and K(q)). Illustrate on the graph from a) by drawing corresponding isocost lines. c) Draw the Cheburashka's expansion path on a graph from a). Derive an expression for the expansion path, i.e., K as a function of L. How does it depend on the wage and the price of kiwi?