Solve the optimality condition for each P equation against M according to the following relation:
Condition for Optimality: ∇M = λ ∇P with respect to C and T.
Note that the ∇is the gradient vector, and λ is a scalar (the Langrange multiplier) so that:
M is the objective function and P is the constraint function.
The solution is obtained by differentiating M and P twice ( that is one time with respect to each variable C and T.
Then finding the values of λ. And finally substituting in the constraint function P and find the values for C and T. (( In this method we assume that ∇P is NOT equal to zero.))
Thus, solve for C and T:
dM/dC = λ (dP/dC) and dM/dT = λ (dP/dT)
M will be the objective function, and P will be the constraint function.
Note The total solution must be repeated three times:
∇M = λ ∇P1
∇M = λ ∇P2
∇M = λ ∇P