Form a Lagragian equation
Q=4(L0.5)(K0.5) subject to wL+rK=C
L=4 (L0.5)(K0.5) −λ(wL+rK−C)
δL/δL=2(L−0.5)(K0.5)−λw=0..........(i)
δL/δK=2 (L0.5)(K−0.5) 5−λr=0...........(ii)
δL/δλ=wL+rK−C=0.................(iii)
Divide equation (i) and (ii)
K/L=w/r and thus K=wL/r and L=rK/w
Replacing the two equation on equation (iii)
w(kr/w)+rK=C thus K∗=C/2r
(K∗)=200/(2∗)5=20
wL+r(wL/r)=C
wL+wL=C
(L∗)=C/2w
(L∗)=200/(2∗)3=33.33
Hence the optimum output is;
Q=4 (33.330.5)(200.5)
Q=4(4.58)(4.5)
Q=104.40 units - Maximum Output.
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