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Business operates on the basis of the production function $Q=25\cdot K^{1/3}\cdot L^{2/3}$ (where $L$ = units of work and $K$ = units of capital).

If the prices of inputs $K$ and $L$ are respectively $3$ euros and $6$ euros per unit, then find :

a) the optimal combination of inputs that the company must occupy to minimize its costs, producing $Q = 600$ production units. What is the minimum production cost?

b) the optimal combination of inputs that the company must occupy to maximize production, if the amount of money available for the purchase of inputs is $450$ euros. What is the maximum possible level of production?

I have done the following:

a) We consider the function $f(K,L)=600\cdot 25\cdot K^{1/3}\cdot L^{2/3}$. Do we apply now Lagrange multipliers?

b) We consider the cost function $C(K,L)=K+L$. Do we apply now Lagrange multipliers? But for which function?

:unsure: