Find ans classify the stationary points for the following functions:
Q= 5㏑L + 2㏑K - 0.1L - 0.4K
If that is
Q = 5L^2 +2K^2 -0.1L -0.4K
then,
get the partial derivatives of Q with respect to L and K, and set those to zero.
Let me express "partial deivative of Q with respect to L" as DQ/DL, so,
DQ/DL = 10L -0.1
0 = 10L -0.1
L = 0.1/10 = 0.01
DQ/DK = 4K -0.4
0 = 4K -0.4
K = 0.4/4 = 0.1
Thus, for those values,
Q = 5(0.01)^2 +2(0.1)^2 -0.1(0.01) -0.4(0.1) = -0.0205
Hence, if the point is expressed in (L,K,Q), then,
(0.01, 0.1, -0.0205)
is the stationary point.
To check if if it is maximum, minimum or a saddle point, we look at the second partial derivatives.
D/DL of DQ/DL = 10 ......positive
D/DK of DQ/DK = 4 .......positive also
Meaning, the stationary point could be a minimum.
Let's chect the mixed partial,
D/DL of DQ/DK = 0
So
(10)(4) -0 = 40 ......positive
Meaning, the stationary point is really a minimum point.