Cost Minimization function with exponent
OK, so the equation is:
X = K^.25 L^.75, w = 2, r = 4
I know that:
Marginal Product of Labour/Marginal product of Kapital = w/r
So I take the derivative with regards to "l" on top and do the same with respect to "k" on bottom and get...
(3/4K^.25 L^-0.25)/(1/4K^-0.75 L^0.25) = 2/4
The 4's on bottom negate each other...and we know that the negative exponent flips to the other side of the dividing line, which gives us:
3K/L = 1/2
Cross multiply, survey says...
6K = L
Now, I know I have to plug this back into the original equation (ie. for every "L" there is 6"k")...when I do that, I get
X = K^0.25 6K^0.75
My question is where do I go from here, and did I do the math right?