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Answer:
The proof is completed below
Step-by-step explanation:
1) Definition of info given
We have the function that we want to maximize given by (1)
(1)
And the constraint is given by
2) Methodology to solve the problem
On this case in order to maximize the function on equation (1) we need to calculate the partial derivates respect to L and K, since we have two variables.
Then we can use the method of Lagrange multipliers and solve a system of equations. Since that is the appropiate method when we want to maximize a function with more than 1 variable.
The final step will be obtain the values K and L that maximizes the function
3) Calculate the partial derivates
Computing the derivates respect to L and K produce this:
4) Apply the method of lagrange multipliers
Using this method we have this system of equations:
And replacing what we got for the partial derivates we got:
(2)
(3)
(4)
Now we can cancel the Lagrange multiplier with equations (2) and (3), dividing these equations:
(4)
And simplyfing equation (4) we got:
(5)
4) Solve for L and K
We can cross multiply equation (5) and we got
And we can set up this last equation equal to 0
(6)
Now we can set up the following system of equations:
(a)
(b)
We can mutltiply the equation (a) by on both sides and add the result to equation (b) and we got:
And we can solve for L on this case:
And now in order to obtain K we can replace the result obtained for L into equations (a) or (b), replacing into equation (a)
With this we have completed the proof.