Determining the maximum value of a multivariable function under 4 inequality constraints.(Math GRE subject test 9768 Q.25)

Question

I know that I should use Lagrange multiplier method, but how with the inequality constraints? could anyone help me please?

Answer 1

The answer should be D.

Often you can solve these types of problems without needing any fancy math.

Looking at $f$ we see that $f$ will be as large as possible when $x$ is as large as possible and when $y$ is as small as possible. The first two inequalities tell us that the maximum $x$ can be is 2 and the smallest $y$ can be is 0. We can see that these satisfy the other two inequalities so we are done.

Answer 2

$f(x,y)=\text{First Term }- \text{ Second Term}$

$f$ will be maximised when $\text{First Term}$ is maximised and $\text{ Second Term}$ is minimised.

Note that first term is maximised when $x=2$ (Because $x \le 2$). Also note that the second term is minimised when $y=0$ (Because $y \ge 0$). Clearly, when $x=2$ and $y=0$, the conditions $x+y \ge 1$ and $y-x \le 0$ are already satisfied.

Therefore, the required value is $f(2,0)=5(2)-4(0)=10$, which is option $\text{D}$.