I've looked through the site and researched but could not find any help on how to solve this: solve the following problem once with substitution and once with the Lagrange method
Max($x_1$* $x_2$) subject to $2x_1$ + $2x_2$ = 20
I'm not really even sure where to begin with the substitution or Lagrange method. Any and all help would be greatly appreciated. If you know the answer please post so I can work backwards and teach myself. Thank you.
Substitution: use the constraint to get $x_2 = 10-x_1$, then replace in the objective function to have $$\max (10-x_1)x_1$$ This is a concave function, so first-order condition is necessary and sufficient. Equate first derivative to zero and find $x_1^*=5$. Replace this in constraint and obtain $x^*_2=5$. Then $(x^*_1,x^*_2)$ is the solution.
Lagrange method: write the Lagrangian $$L= x_1x_2 - \lambda(x_1+x_2 -10)$$ The problem can be solved using first-order conditions (I am skipping details, but it can recasted as concave programming). Equate the derivatives of the Lagrangian w.r.t. to $x_1,x_2$ to zero and add the constraint to obtain a system of three equations in the three unknowns $x_1,x_2, \lambda$: $$x_2-\lambda =0\\ x_1 - \lambda = 0\\ x_1+x_2=10$$ and solve this to get $x^*_1=x^*_2=\lambda^*=5$.