1.1
Only the formula of chain rule: 2p
Differentiation without using the chain rule: 3p
If chain rule is used but it is not fully apparent how -> -1p
1.2
The condition for partial of F w.r.t y not vanishing: +3p
The formula for condition for partial of F w.r.t y not vanishing in this particular case: +1p
The "branch" x\neq 0 of the condition: +1p
The other "branch" y\neq -2x^2 of the condition: +1p
If only formula for the derivative of y: +2p
2.1
Revenue +1p, cost +1p, profit +1p, optimization problem +1p (note: non-negativity constraints not required)
Note: the mistakes are not cumulative, e.g if wrong cost fucntion then there is no penalty from using it in the profit function
Note: here you are not supposed to use symmetry (-1p if everything formulated such that x_1=x_2)
2.2
FOC for the problem without any symmetry etc arguments
note 1: -1p if everything formulated such that x_1=x_2
note 2: If othwerwise correct but some a mistake in the problem formulation: max 2 points
2.3
Only a solution without any arguments: -1p
If otherwise correct but some a mistake in the problem formulation max 2 points, exception: if the FOCs are solved for a more difficult problem than the correct one, max 3p
3.1
Correct FOCs +3p
Correct solution from the FOCs +3p
Note 1: solution involves Lagrange multiplier, if its missing -> -1p
Note 1: if multiple solutions are given, amonn which only one is correct but not indicated which one -> -1p
3.2
+1p/correct Hessian & +1p/correct definitiness
Note: deduction(s) if it Hessian(s) is(are) correct but unclear how they are obtained
Bordered Hessian +2p (this was not what was asked nor is needed)
3.3
Mistake in some Hessians but correct explanations: 1p
Mentioning that the global optimality +1p
Relating definiteness to concavity/convexity +1p