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